Analysis of influence of fibre type and orientation on dynamic properties of polymer laminates for evaluation of their damping and self-heating

2.1 Self-heating effect

The self-heating phenomenon is generally caused by the presence of damping in materials. Damping in mechanical systems is understood as an irreversible transition of mechanical energy into other forms of energy, mostly heat. Considering the fact that most polymers used as a matrix of composites are characterized by a low heat transfer coefficient, the generated heat is stored in the structure causing a rise in temperature. In the case when the rate of thermal energy is greater than the one transferred to surrounding media in conduction, convection or radiation mechanisms, the rise of the temperature may initiate a thermal fatigue process, which initiates damage evolution and ultimately the destruction of a structure [36].

2.2 Mechanical- and chemical-related degradation of composites

Considering the three-phase nature of self-heating temperature growth [36], the associated degradation process could be explained by two hypotheses. The first hypothesis describes the possible temperature non-linear changes due to the effect of residual cross-linking processes on mechanical properties. The second hypothesis relates the temperature growth due to the self-heating to the initiation of micro-cracks in the second quasi-linear phase of the thermal fatigue.

The results of an additional analysis [32], [33] show that residual cross-linking during the self-heating process is expected to be led by reactions between hydroxyl groups and epoxides in the investigated specimens. Considering the analysis of Raman spectra presented in [37] the evolution of the peak intensity of epoxy ring breathing vibration at 1256 cm^-1 [37] and the evaluation of the residual cross-linking could be analysed. This could be an indicator for the existence of critical temperature of the self-heating, which characterizes the initiation of mechanical micro-damage sites caused by the temperature as well as for the presence of the relation between the heating rate and dynamic properties of the investigated materials.

The second hypothesis results from the first one: the character of the self-heating temperature changes can be accounted for the changes of stiffness due to the residual cross-linking and the associated changes of damping. Following the second hypothesis, the damping should increase due to the initiation of micro-cracks and associated energy dissipating via micro-friction [18]. However, the damping characteristics under elevated temperatures could be highly non-linear, which may cause the changes of the evolution character of the self-heating temperature curve.

3.1 Thermoviscoelasticity and relation between self-heating and damping

The linear thermoviscoelastic materials could be described following the Boltzmann superposition principle by taking into consideration a temperature dependence given by the equation

where σ(t) and ε(t) are the stress and strain histories, respectively, T is the temperature, E₀(T) is the temperature- dependent instantaneous modulus of elasticity, τ is the relaxation time and R(t-τ, T) is the temperature-dependent relaxation kernel.

Considering the cyclic loading of the structures, the stress and strain relations are given as follows:

where σ₀ and ε₀ are the stress and strain amplitudes, respectively, ω is the angular velocity and δ is a phase lag.

Applying the Fourier transform to (1), it is possible to present an equation of thermoviscoelasticity using complex parameters in the frequency domain:

where R*(ω, T) is the complex relaxation kernel.

The complex modulus could be presented as

where

are the storage and loss moduli, respectively. Additionally, the dynamic moduli (6) and (7) are in relation following the formula

which describes the tangent of a loss angle depending on the excitation frequency and the temperature.

It should be noted that the self-heating effect occurring during cyclic loading which causes an increase in local temperature has the same nature as an evolution of the local damping properties of the material considered in the same conditions (i.e. a frequency and temperature variation). Following the theoretical model of local damping of fibrous composites presented in [4], the damping can be described originating from the energy dissipated during cyclic dynamic loading. In the general case, the dissipation energy integrated over a cycle, ΔW_d , takes the following form:

where P is the period of loading; X is the vector of Cartesian coordinates x, y and z; and ψ is the specific damping capacity.

The latter parameter is connected with the loss factor η and tangent of loss angle as follows [12]:

Similarly, the dissipation energy could be defined for the hysteretic heat generation problem [3]. The dissipation energy function for the heat generation, ΔW_h , could be given by

Thus, ΔW_d ≡ΔW_h due to the following relation (cf. [38]):

where H(ω, T) is an exothermal energy dissipation function. The total amount of dissipated energy consists of parts of plastic deformations [39], electromagnetic emission [40], etc.; however, the greatest part of a total amount of energy is dissipated in the form of heat [39].

In order to determine the behaviour of the complex parameters at variable excitation frequencies, temperature values, heating rates and a fibre type and orientation, a series of experimental studies were carried out using DMA. The fundamental kinetic relations based on the modified Arrhenius law are given in the next subsection.

3.2 Kinetic relations

The determination of dynamic parameters in order to fulfil (12) can be precisely performed using the DMA. In order to characterize dependences between E′ and E″ with the excitation frequencies, temperature, heating rates and a fibre type and orientation, first the glass-transition temperature T_g should be obtained in each particular case. The determination of T_g from DMA measurements is possible based on different dynamic parameters: as an onset of E′(T) drop, as a peak of tan δ(T) or as a peak of E″(T), which is the most commonly used and suggested by the European Standard EN ISO 6721-1 [41].

Depending on the changes of the excitation frequency f, T_g changes its values following the mechanochemical kinetic relation – the Arrhenius relationship, which describes the amount of the molecular energy needed for the transition – in the considered study the glass transition:

where f₀ is the empirically determined pre-exponential factor, E_a is the activation energy needed for the transition, R is the universal gas constant (8.314472 J/K·mol) and T_r is the reference temperature (in the present study T_r =T_g ).

The pre-exponential factor can be determined by the analysis of kinetics of the reaction and described as a relationship between the chemical reaction rate and temperature. The determined glass-transition temperatures were presented against the excitation frequency as the Arrhenius plot: 1/T_g vs. log f. The linear regression of the obtained curve is calculated, and the slope describes the activation energy. Equation (13) could be modified in order to obtain the shift factors a_T for the application of the TTS principle. Following the first hypothesis presented above, this relation was modified by introducing the heating rate factor, which describes additionally the influence of a heating rate on the shift factors [3]:

where T_r (β_T ) is the reference temperature for a given heating rate β.

Following the TTS principle, which assumes that the value of a given dynamic parameter at the reference temperature T_r could be related to another temperature T by multiplying the frequency by the shift factor for the temperature T:

where •( ) denotes any given dynamic parameter.

Having the relation between shift factors and excitation frequency, the construction of master curves is possible. The curves of isothermal dynamic parameters are presented as the frequency domain vs. appropriate values of a_T determined for the single reference temperature.

4.1 Specimens preparation

Three main types of specimens were analysed in the experiments: glass fibre (GFRP) and carbon fibre (CFRP) reinforced epoxy resin as well as pure epoxy. The first two were manufactured in an autoclave process using unidirectional-reinforced epoxy preforms. The specimens with the length of 40±0.2 mm, width of 10±0.5 mm and thickness of 5±0.5 mm were cut from the manufactured 20-ply 0°-reinforced plates and stacked at the following angles: 0°, 45° and 90° with respect to the global coordinate system of plates. The pure epoxy resin specimens were manufactured using pure epoxy preforms in autoclave technique as well, and used as a reference in the following analyses. The parameters controlling the manufacturing process were set according to the recommendations of the manufacturer of the preform tapes.

4.2 Measurements

The tests were carried out on the TA DMA Q800 Dynamic Mechanical Analyser (TA Instruments, New Castle, DE, USA). The specimens were fixed in the claps of DMA with the constant force moment of 1 N m. The single cantilever deflection mode was selected; the load was realized with the constant deflection amplitude of 80 μm.

In order to consider the influence of the heating rate on the measured dynamic parameters, the tests were performed in a single-frequency excitation mode for the following frequencies: 1, 10 and 20 Hz. The temperature range was set to 293–493 K. In order to stabilize the temperature in the chamber, the isothermal mode at the beginning of each test was set to 3 min. The tests were performed for three different heating rates: 3, 8 and 15 K/min. For each unique case, the three specimens were tested, which gave 81 tests for each reinforced laminate (considering different excitation frequencies, heating rates and fibre orientations) and 27 tests for the specimens made of pure epoxy (variable excitation frequencies and heating rates).

During the tests the following functions were measured: E′(T), E″(T) and tan δ(T). The T_g values were evaluated based on the peak of E″(T).

5.1 Estimation of the activation energy

Considering the fact that the absolute accuracy of the DMA temperature readout depends on the temperature lag between the dynamic mechanical analyser temperature sensor and the specimen, the obtained values of T_g should be corrected by taking into account at least the occurring convection phenomena especially during the measurements of dynamic parameters at various heating rates [23]. For this purpose, a simple model in the Matlab/Simulink^® (MathWorks Inc., Natick, MA, USA) environment was prepared (see Figure 1). The mechanical and thermal properties were assumed for GFRP, CFRP and pure epoxy, respectively. These properties are presented in Table 1. The exemplary results of taking into account the convection following the model presented in Figure 1 are given in Figure 2.

Figure 1:

The Matlab/Simulink model for considering of convection phenomena.

Figure 2:

Exemplary results for temperature lag occurring from convection phenomena for the GFRP specimen with the heating rate of 8 K/min.

Following the results obtained from the developed model (Figure 1), the determined values of T_g were corrected. These values for the above-described sets of parameters are presented in Table 2. The misestimation of T_g values due to the convection lag is presented in Table 3. The activation energies for these cases were calculated from the slope of linear regression of T_g (f) functions from the Arrhenius plots and are summarized in Table 4. The Arrhenius plots for the GFRP and CFRP laminates are presented in Figure 3. In this figure, the markers denote measurement data for the specimens with fibre orientations of 0°, 45° and 90°, respectively, while the lines present the regression of measurement data.

Figure 3:

Arrhenius plots for GFRP and CFRP.

The same calculations were made for the pure epoxy specimens in order to evaluate the influence of the fibre presence and type on the dynamic properties, local damping and self-heating. The Arrhenius plot for the pure epoxy for various heating rates is presented in Figure 4, while the T_g and E_a values are presented in Table 5. The obtained values of T_g and E_a are in good agreement with typical values for epoxies [42] and FRP composites [3], [10], [43].

Figure 4:

Arrhenius plot for pure epoxy.

Following the results of the activation energy analyses for different considered materials, it can be observed that the reinforcement presence has a great influence on the necessary energy amount for transition initiation. E_a depends on the mechanical properties, and hence the presence of the reinforcement can increase the values of E_a (cf. Tables 4 and 5). From the comparison of E_a of the investigated laminates, it could be noticed that the values of E_a for GFRP are always higher than those for CFRP regardless of the variation of β and θ. Taking into account the fact that the values of the dynamic properties of CFRP are higher than those for GFRP (which is discussed in the next subsection), the values of the activation energy reveal an inverse relation. This fact has an explanation in the thermal properties of materials of the reinforcement. The glass fibre is generally a thermal insulator, while the carbon fibre is a heat conductor. Therefore, for CFRP the T_g point could be reached faster, which has a direct influence on the damping capabilities and self-heating. As was previously stated [see (9) and (11)], the dynamic properties are in direct dependence with the dissipation energy related to damping and self-heating, that is, depending on the heat conduction coefficients of the material of reinforcement the obtained values of dynamic properties may behave in a different manner.

One can also observe that the values of T_g and E_a are sometimes higher for the heating rate of 8 K/min than the values for other considered heating rates. This is probably resulted from a non-linear relation of T_g and cross-linking ratio, which was proven experimentally by DMA tests [30] and by Raman spectroscopy tests [32].

The obtained values of T_g confirmed previous results [3] of the linear relation between T_g and β. Considering (14), it is necessary to define the reference heating rate β_r . Following the requirements of the European Standard EN ISO 6721-1, it was assumed that β_r =3 K/min.

5.2 Frequency shift factors and master curves

For the determination of frequency shift factors a_T , the reference temperatures must be assumed for each investigated case of a heating rate. In the present study, the value of T_r was assumed as the average of glass transition temperatures T̅_g for the reference heating rate β_r . Similarly, T̅_g were determined for other heating rates β≠β_r , which are tabulated in Table 6. Shift factors could be determined from the relation of log a_T -(T-T_r ) for a given heating rate.

Following the reference temperatures presented in Table 6, the horizontal shift factors a_T (β) were determined. Then, the master curves for dynamic moduli and the tangent loss factor could be constructed by transforming the temperature scans to the frequency domain. Using the calculated shift factors, the isothermal frequency-dependent curves were shifted against the reference temperature [see (15)] for each value of the heating rate. In order to depict the influence of the fibre presence and its type, the exemplary master curves for storage and loss moduli and for the tangent loss factor are presented in Figures 5–7, respectively. In these figures, the investigated materials are marked by different colours, while the type of curves denotes the heating rates. For the laminates, the fibre orientation was 0°.

Figure 5:

Master curves of the storage modulus for various materials and heating rates.

Figure 6:

Master curves of the loss modulus for various materials and heating rates.

Figure 7:

Master curves of the tangent loss factor for various materials and heating rates.

The master curves of the storage modulus presented in Figure 5 confirmed the previously stated hypothesis that the thermal properties of the material of reinforcement have an influence on the evolution of the damping properties and the self-heating effect. The results also indicate that the presence of the reinforcement increases much the loss modulus (see Figure 6), which is directly related to the amount of the self-heating temperature. In the case of damping it could be noticed that the fibre presence decreases the damping properties of the material at the T_g point. As can be seen in Figure 7, the pure epoxy at this point has the values of tanδ near 1, while the reinforced laminates reveal much smaller values. However, at typical operational temperatures, the values of tangent loss factors for all investigated materials are comparable. The heating rate influences the dynamic properties of materials with different fibre orientations; however, no relations between the fibre orientation and the heating rate could be observed.

The next study was concerned with the determination of the influence of the fibre orientation on the damping capabilities and self-heating. The master curves for different materials with different fibre orientations are presented in Figures 8–10. In these figures, the investigated materials are marked by different colours, while the type of curves denotes the fibre orientation. The master curves were plotted for the heating rate of 8 K/min.

Figure 8:

Master curves of the storage modulus for various materials and fibre orientation.

Figure 9:

Master curves of the loss modulus for various materials and fibre orientation.

Figure 10:

Master curves of the tangent loss factor for various materials and fibre orientation.

While considering the results presented in Figure 8, the resulting master curves of a storage modulus reveal such a tendency that the lower the angle of fibre orientation, the higher the value of a storage modulus. This observation is directly connected with the presence and orientation of reinforcement, since the storage modulus represents an elastic response of the analysed structures. The values of a master curve of E′ for pure epoxy additionally confirm these statements. Similar results and conclusions were presented by Berthelot and Sefrani [11]. Similar character is also observed for master curves for the loss modulus presented in Figure 9. Analysing the loss factors presented in Figure 10, one can observe that the lowest values were obtained for 0°, higher values for 45° and the highest ones for 90° of fibre orientation both for GFRP and CFRP structures, which is in agreement with experimental results and results obtained from theoretical models presented by Berthelot et al. [14].

Analysing Figure 8, one can notice that the storage modulus for CFRP is higher than GFRP for the fibre orientation 0°, while in the case of loss modulus for the same pair of curves (see Figure 9), the values for GFRP are higher than those for CFRP. This fact proves that the dissipation energy and, thus, the heating energy also depend on the conductivity of the reinforcement, that is, considering that in the carbon fibre – characterized by good thermal conductivity – the heat exchange between the polymer matrix and fibre is better and hence the heat transfer to the surrounding medium is possible.

The presented approach of determination of previously described relations has several limitations. Due to the complex nature of thermal processes occurred in a DMA chamber (misestimation of true temperature due to convection and position of the temperature sensor in a chamber, thermal anisotropy of FRP composites, etc.), exact values of temperature and its evolution over time need to be analysed in more detail; however, the presented approach of temperature correction (see Section 5.1 and Figure 1) partially solves this problem. Another limitation is resulted from the degrees of cross-linking of the tested materials, which are not known in this study. Taking this parameter into consideration will allow for better parametrization of the applied model (14), which is planned in further studies on similar materials with various degrees of cross-linking using both DMA and Raman spectroscopy.