In this paper, an acoustic fatigue model is proposed to obtain the residual strength of plain weave C/SiC composite plate subjected to acoustic loading. The constant amplitude normalized residual strength model and the S-N curves are used in new model, in which the probability density function of stress peaks are adopted to evaluate the effects of random loading on the degradation of residual strength. The degradation of residual strength in a time increment is calculated via an integration of the decrease of residual strength due to the stress peaks in the definitional domain of the peak probability density function. The evolutions of the residual strength under acoustic loading are determined via a double integral method. The residual strengths of C/SiC composite plate predicted by proposed model are in good agreement with the experimental results.

Advanced composite materials with high specific strength and high specific stiffness have been widely used in many fields [1]. Under strong acoustic loading, the surface of composite material plate will produce large deflection displacement. In the meanwhile, the performance of the composite material keeps decreasing, and the damage keeps accumulating, eventually leading to the failure damage. Due to the anisotropy, brittle and non-homogeneity of the composite material, the failure modes are degumming, delamination, fracture and so on. These characteristics lead to the complexity of fatigue model. Therefore, to predict the life of C/SiC composites under acoustic loading precisely, the selection of damage variables and the corresponding failure criteria are crucial.

The common methods are time domain method and frequency domain method. The time domain method is mainly based on cyclic counting technology, and it can extract the stress state and related cycle number from the random response. Mei and Chen [2] used the equivalent linearization method to study the nonlinear random response of arbitrarily shaped composite plates in the high temperature environment. Vaicaitis [3–5] combined Galerkin method and Monte-Carlo method in time domain to analyze the nonlinear response of composite plates under thermal acoustic loading. Dhainaut [6] used finite element model and numerical integration to study the random response and fatigue life of composite plates under Gaussian and non-Gaussian excitation. Based on Von Karman's large deflection equation and the theory of laminates, the motion equation of composite laminates was deduced, and the time domain method was used to sample directly and count the stress time history of the structure. However, the disadvantage of time domain method is slow calculation efficiency. The frequency domain method makes up for this liminit. The frequency domain method uses probability density function (PDF) and power spectral density (PSD) to describe the random response [7,8]. However, the liminit of the frequency-domain method is that the selection of loading sequence has a great impact on the result of damage accumulation, so it is not suitable for direct use. For regular cyclic loadings, the nonlinear cumulative damage models of composite were proposed based on the concept of residual strength and have been widely adopted [9,10].

In both of the above situations, the fatigue life of constant amplitude will be determined by the Palmgren–Miner rule [11,12], which is independent for the loading sequence. However, the study in this paper is about random loading, the prediction results of the simple linear cumulative damage model are not accurate [13]. Therefore, an acoustic fatigue model is established, where the concept of residual strength [14–20] is used as the damage variable to represent the damage of composite. Residual strength model combined with frequency domain analysis are used to predict the failure of composite plate under acoustic loading in this paper. The peak value of stress response is calculated by ABAQUS. The probability density function (PDF) of stress peak adopted in this paper could eliminate the influence of loading sequence on damage accumulation, and the curve of the residual strength distribution of the plate is obtained by the acoustic model proposed in this paper. The accuracy of the model was verified by comparing experiment and simulation results. To establish and verify the acoustic fatigue model, the experiments are quite necessary, including the constant amplitude and acoustic fatigue experiments. In the constant amplitude fatigue experiment, the residual strength curves of the braided plate composite under various stress ratios are obtained. In the acoustic fatigue experiments, the strong acoustic loading is applied to the surface of the plate, and the plate is subjected to acoustic fatigue test. After the acoustic fatigue experiment, the plates were segmented, and the specimens were subjected to the residual strength test, and the residual strength distribution diagram of the plates was obtained. The experimental results are compared with the acoustic fatigue model.

2Acoustic fatigue modelThe calculation process of the proposed fatigue model is shown in Fig.1. The constant amplitude normalized residual strength model, probability density function of stress peak, and the S-N curves are adopted to obtain the random residual strength model. Firstly, the degradation of residual strength in a time increment is calculated via an integration of the decrease of normalized residual strength due to the constant amplitude fatigue loading. Secondly, the probability density function of stress peak is adopted, the assumption in this paper is that the acoustic loading is ergodic, which means all the peak stress appear in the unit time increment, when the composite is subjected to acoustic loading, the stress peak in the time increment is random, so the probability density function of stress peak is quite important for the characteristic the effect of acoustic loading on strength degradation in time increment, the S-N curves is also needed for the strength degradation due to the different stress peak. Lastly, the evolutions of the residual strength under random loading are determined via a double integral method. The evolutions of random residual strength can be written as Eq. (1):

Where Sp is stress peak, pSp is probability density function of stress peak, ΔSrcSp,nΔn is strength degradation via the cycle number due to the constant amplitude fatigue loading. ΔSrrnΔn is random strength degradation via cycle number due to acoustic loading. To obtain the evolutions of the random residual strength, the following theories are necessary.2.1Normalized residual strength model under constant amplitude fatigue loadingBased on the model of residual strength under constant amplitude alternating loading and combining with S-N curve of composite materials and residual strength test data, the evolution of residual strength degradation of thin-wall structure under acoustic loading is calculated. The normalized residual strength model under constant amplitude fatigue loading can be described by a two-parameter nonlinear theoretical model, as followed:

It needs to be pointed that r is the normalized residual strength, and t is the normalized number of cyclic, α and β in Eq. (2) are the parameters of curve fitting.

In Eq. (3)Sr is residual strength, S0 is static strength, and Smax is peak stress. N in Eq. (4) is at applied stress state.

2.2Equivalent residual strength under variable amplitude loadingThe plain weave C/SiC composite plate will be subjected to alternating amplitude loading under acoustic loading in the time increment as shown in Eq. (1), its residual strength evolution will change with the stress state, and the influence of the loading sequence should also be considered. Therefore, the equal residual strength degradation under different stress states should be adopted to obtain the residual strength in the time increment.

As shown in Fig. 2, the residual strength evolution curves under the three stress states are presented, Sp and R represent the value of stress peak and stress ratio, respectively. The concept of equivalent cycle number neq should be introduced, when the stress state of the composite changes.

Under constant amplitude alternating loading Sp1 and ratio R1, the residual strength S0 turns to Sr1, after n1 cyclic. Before the unit bears the alternating load of constant amplitude Sp2 and ratio R2, it needs to calculate the equivalent cycle neq1 corresponding to Sr1 in the state of Sp1, R1. Then, the unit experiences the alternating load of constant amplitude Sp2 and ratio R2, after n2 cyclic. It is equivalent that this unit bears loading conditions Sp2, R2 after neq1+ n2 cyclic with residual strength from S0 to Sr2. Similarly, after calculating the equivalent cyclic neq2 under the loading conditions Sp3, R3, the residual strength Sr3 can be obtained.

2.3Probability density function of response peakAt room temperature, narrow-band Gaussian white acoustic is applied to the plain weave C/SiC composite plate in this paper.

In the narrow-band approximation model, the stress peak of the response is considered to be equal to the amplitude, and both obey Rayleigh distribution.

Then the probability density function of response peak is:

Where, RMS is the stress root mean square value of the unit response, calculated by ABAQUS; Sp is stress peak. Based on the above model, the number of cycles per RMS corresponding to different peaks can be obtained by integration. Taking RMS=110.18MPa as an example, Fig. 3 shows the number of cycles corresponding to a peak value of 0–330.54MPa. Here the stress ratio of all cycles R=−1, which is also determined by the nature of Gaussian white acoustic itself.2.4Evolution of random residual strength for C/SiC compositeAfter the introduction of the fatigue and random theories, the random residual strength model proposed in this paper can be established. As shown in left of Fig. 4, when unit time increment (dt) increases from ti to ti+1, then residual strength decrease dSr reduce from Sri+1. to Sri. During the time increment, the different stress peaks appear according to the probability density function of response peak shown in the right of Fig. 4.

The random residual strength with fatigue cycle is presented in Eq. (1). Then the evolutions of the residual strength with time are determined via a double integral method.

First of all, a diﬀ ;erential of Eq. (2) is performed to get dSrcdn

Then, by combining the Eqs. (2), (3), (5) and (6), the Eq. (1) could become

Then evolution of random residual strength could be obtained by adouble integral method.

3Experiments3.1Materials properties experimentsThe 2-D orthogonal plain weave C/SiC ceramic matrix composite material specimens used in the experiment were manufactured by the ultrahigh temperature composite laboratory in Northwestern Polytechnical University using chemical vapor infiltration process technology. The main component is T300 carbon fiber and its matrix is SiC. The ratio of warp yarn to weft is 1:1 [21].

The density of the material is about 2g/cm3, void fraction is about 10–15%, fiber volume content is about 40–45%, and matrix volume content is about 40–50%. The properties of carbon fiber and SiC matrix [22] are shown in Table 1.

According to the experimental standard GBT 1447-2005 and GJB 6476-2008, three kinds of materials fatigue experiments were carried out in room temperature environment, including static tensile/compression strength test along the fiber direction, the tension-tension fatigue (T-T) test, the compressive-compressive fatigue (C-C) test and their residual strength test.

The static experimental results are introduced in this paper, which were obtained from the literature [23]. Static tensile and compression tests were carried out at room temperature. The average value of tensile static strength of the material is 221.6MPa. The average value of compressive static strength is 348.29MPa.

3.1.1S-N curve experimentFor 0/90° braided composites, the X and Y direction have the same mechanical properties, and ignore other strength variation influence. The residual strengths of XT and XC are studied in this paper.

It was found [23] that for the residual strength of composite materials under constant amplitude loading, the residual strength evolution law of different strength can be normalization at the same stress ratio, and the influence of the variation of nonlinear parameter on the law can be neglected. The equation of residual strength evolution is shown in Section 3.1 of this paper. The assumption is that under the same fatigue mode (tension-tension, compress-compress, or tension-compress), such as the tension-tension fatigue loading, the influence of the nonlinear parameters corresponding to different stress ratios on the evolution of the residual strength is also ignored. Therefore, in the case of the tension-tension fatigue loading, the residual strength evolution law of the stress ratio 0≤R<1 and varied strength peak can be obtained by using the S-N curves and determined nonlinear parameters, which is at the conditions of R=0.1 and strength peak is σmax1. Similarly, in the case of compressive-compressive fatigue loading, the residual strength evolution law of the stress ratio R>1 and variation strength peak can be obtained by using the S-N curves and determined nonlinear parameters, which is at the conditions of R=10 and strength peak is σmax2. Then, the nonlinear parameters to be determined in this paper are shown in Table 2.

Nonlinear parameters for Eq. (2).

Nonlinear parameters | Residual strength | Stress Ratio | Strength peak |
---|---|---|---|

α(T), β(T) | XT | 0.1 | σmax1 |

α(C), β(C) | XC | 10 | σmax2 |

C/SiC composite was subjected to constant amplitude alternating load fatigue life test at room temperature, and the test equipment was Instron-8801 fatigue testing machine, as shown in Fig. 5.

In the tension-tension fatigue test, the stress ratio R=0.1, the frequency f=25HZ, the life of the material is more than 106 times when the stress level is below 0.87, and the life starts to be less than 106 times when the stress level is higher than 0.89, here, the stress level (LV) is defined as stress divided by static strength.

The experimental data and the fitted S-N curve are given by Eq. (8) in Fig. 6,

According to the fitting curve, the corresponding stress level of 106 life cycles is 0.896.

In the pressure-pressure fatigue test, the stress ratio R=10, the frequency f=25HZ, the life of material is more than 106 times when the stress level is below 0.85, and the life starts to be less than 106 times when the stress level is higher than 0.87.

The experimental data and the fitted S-N curve are given by Eq. (9) in Fig. 6,

According to the fitting curve, the corresponding stress level of 106 life cycles is 0.86.

In the Eqs. (8) and (9), S is the stress peak, N is the number of cyclic, σULT is the tensile static strength, and σULC is the compressive static strength. It can be seen from the above fatigue data, the tension-tension and compressive-compressive fatigue strength of plain weave C/SiC composites are higher than 85% of the corresponding static strength.

3.1.2Residual strength experimentsThe residual strength of C/SiC composites under constant amplitude loading was tested at room temperature. The test conditions of tension-tension fatigue residual strength were: The stress level LV=0.9, the stress ratio R=0.1, and the frequency f=25HZ. The test conditions of compressive-compressive fatigue residual strength were: the stress level LV=0.87, the stress ratio R=10 and the frequency f=25HZ.

Residual strength test is a kind of static tensile/compression test of the composite after a certain number of cycles loading. Considering that the strength of composite material is generally not significantly degraded within 104 at room temperature, the number of cycles loading selected for tension-tension residual strength test are 105, 2×105 and 3×105, respectively.

The normalized residual strength evolution equation was used to describe the T-T and C-C residual strength degradation, as shown in Fig. 7.

It can be seen from the experimental data that the distribution of residual strength is not regular due to the dispersion of the strength of the material itself. Then, in this paper, the normal temperature tension-tension and pressure-pressure experimental data are mixed together, expressed in the form of normalization shown in Table 3, and the evolution law is obtained.

3.2Structural experimentsIn this section, the effect of acoustic loading on the residual strength of the C/SiC plain weave composite plate is studied.

3.2.1Acoustic fatigue experimentIn the acoustic test, C/SiC composite thin-wall structure is connected to the traveling wave tube of the test system through the test fixture. The flexible gasket is set between test specimen and fixture. The mounting form is compression mode. Experimental conditions: the acoustic test in this project will adopt the equal bandwidth acoustic loading spectrum, and control accuracy is required to be ±1.5dB. The test was carried out at room temperature with an acoustic loading pressure level of 160dB and a duration of 300s.

The typical thin-wall structure test under acoustic loading is completed in the acoustic device of traveling-wave tube at room temperature, as shown in Fig. 8. The traveling-wave tube system mainly includes sound source system, test section, acoustic control system, acquisition system and field monitoring system.

3.2.2Residual strength testIt is difficult to use the whole plate to obtain the residual strength of the thin-wall structure. The reasonable solution is to cut and extract from representative locations for testing.

The residual strength distribution verification experiment was carried on the dumbbell type test piece, as shown in Fig. 9, which was intercepted from the typical thin-wall structure after the acoustic test of specified magnitude and time. The residual strength specimen is cut and extracted at the corresponding position of the plate, and the number is shown in the figure. Since the response is symmetrical between the two center lines of the plate, the specimen with the same number has the same residual strength status.

The residual strength test of rectangular plate under static tensile load is carried out by Zwick/z100 universal mechanical testing machine. There are 31 tensile residual strength specimens.

Fig. 12 shows the distribution of the mean value of residual strength at the symmetric position in the thin-walled structure, and the tendency of the mean value of residual strength in the Y direction changing with the X-axis coordinates.

4Results and discussionBased on the proposed acoustic fatigue model, the residual strength evolution of C/SiC composite plate under acoustic loading is simulated. Firstly, the random model of composite is established in ABAQUS. Stress root mean square values (RMS) of all unit elements are recorded for the next calculation. The FE model and Stress root mean square values are shown in top of Fig. 10.

The whole model is under sliding boundary conditions, which is consistent with the experimental condition. The sliding boundary conditions are that the structure perpendicular to the X-axis can slide along the X-axis, while the structure perpendicular to the Y-axis can slide along the Y-axis. The distribution of residual strength of structure can be seen in the bottom of Fig. 10. Based on the Eq. (7), the residual strength curves of C/SiC composite with stress root mean square value (RMS) of 67, 55 and 40MPa are calculated, respectively, which could be seen in Fig. 11. The bigger the RMS value is, the faster the strength decreases.

As shown in Fig. 10, the RMS in the X and Y directions of the plate edge is the smallest, and the RMS is the largest in the central position. The residual strength distribution shows a tendency of decreasing from the edge to the area near to the central position, and in the area near to the central positon, the residual strength increases, but the residual strength reaches the minimum value in the central point, which can be seen in Fig. 12. After comparison between simulation results and experimental data, the tendency of calculation results of residual strength distribution is consistent with the experimental results, and the comparison curves is shown in Fig. 12.

5ConclusionIn this paper, a residual strength model for C/SiC composite under acoustic loading is established. The evolutions of the residual strength of composite under random loading is obtained by using normalized residual strength model of composite under constant amplitude loading, S-N curves and the probability density function of stress peak. To accomplish and verify the proposed model, constant amplitude fatigue experiment and residual strength experiment of C/SiC material are carried out, besides the acoustic loading test and residual strength test of C/SiC plate are also carried out. The residual strengths of C/SiC composite under acoustic loading calculated by the proposed model show good agreement with experimental results.

The acoustic testing device is designed to accomplish the acoustic fatigue experiments of C/SiC composite plate at room temperature, and the acoustic fatigue properties of the C/SiC composite plate subjected to160dB acoustic loading is obtained. As shown in experiments, the residual strength of C/SiC composite plate decreases from the edge of plate, and then increases a little from the edge to the central line in X direction, and in the central line position, the residual strength reaches the minimum value.

In numerical modelling, the tendency of residual strength distribution of C/SiC composite plate is basically consistent with the experimental results. By adopting the probability density function of stress peak, all the random cycle loadings in the unit time increment could be considered in the numerical calculation. The effects of loading sequence on residual strength of C/SiC composite could be ignored, and the acoustic fatigue properties of C/SiC composite could by studied by the proposed model in this paper.

Conflict of interestWe declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Acoustic fatigue properties investigation of plain weave C/SiC composite plate”.

This study is supported by the National Natural Science Foundation of China (NO. 11402064, 11402028 and 11172046).

*et al*.