Mechanical property distribution of composites considering cured ply thickness variability

Variability is an essential feature of composites and directly affects the application of composite structures. Only a few studies have been conducted on the mechanical properties of composites considering cured ply thickness variability. This paper analyses the phenomena of cured ply thickness and the mechanical properties of materials obeying normal distribution after normalization. The distribution function and probability density function of the mechanical properties of composites considering the variability of cured ply thickness are derived. The correctness of the derived distributions is verified by Monte Carlo simulation, which compares the distribution of mechanical properties of composites after normalization with that considering the variability of cured ply thickness. The results showed that after considering the cured ply thickness variability, the mechanical properties showed a skewed distribution. At the lower percentile of the distribution, the mechanical properties are low compared to the normalized values, which reduces the permissible values of the composites.


Introduction
The mechanical properties of composites are fundamental to the design, analysis, and validation of composite structures [1,2].However, there is a natural variability in the properties of composites.Composite tensile and compressive mechanical property test data are often processed using normalization (normalisation) methods to obtain equivalent data for a specific fibre volume content [3].This processing usually significantly reduces the dispersion of the test data and improves the baseline values.Since the cured ply thickness (CPT) of a composite is negatively correlated with the fibre volume content, in engineering practice, it is common to regularise to a specific CPT (representing a specific fibre volume content).The determined CPT facilitates composite structural design and strength analysis using conventional deterministic methods.However, methods for characterising the mechanical properties of materials have developed considerably [4][5][6][7][8].And the values of mechanical properties at specific fibre volume content obtained by the normalization method and the combined environmental conditions method have been obtained with excellent accuracy and reproducibility.
However, the actual composite structures often have variability in the thickness of the cured ply, which makes the measured values of the mechanical properties different from the normalized nominal values.In practical tests, information on the CPT of the composite structure at the strain measurement point is often not easily and accurately obtained.This leads to the fact that the effect of CPT variability must be considered when determining the mechanical property allowable values of actual materials.Therefore, the study of the distribution characteristics of mechanical properties of composite materials, taking into account the CPT variability, is of great significance to further improve the reliability of the permissible values.
This paper is based on the premise that the tensile and compressive mechanical properties of composites are positively correlated with fibre volume content and negatively correlated with CPT.The common cases in which the mechanical properties and CPT of composites obey normal distribution for a specific fibre volume content are investigated.The cumulative distribution function and the probability density function of the mechanical property distribution of composites considering the variability of CPT were derived and verified by simulation using the Monte Carlo method.Numerical simulations were carried out to compare the normalized mechanical property distributions with those considering the CPT variability.

Derivation of formulae
In engineering practice, the original mechanical property values of composites obtained from testing are usually normalized in the following way.
where T is the thickness of the specimen cured ply.t is the specified thickness of the cured ply, which corresponds to a specific fibre volume content, and X is the normalized mechanical property value.It should be emphasised that the raw mechanical properties Z , the normalized mechanical properties X and the specimen CPT T are all random variables.The dimensionless thickness Y is the ratio of the thickness of the single layer of the specimen T to the specified thickness of the single layer t , as shown in Equation (2).
Substituting Equation (2) into Equation ( 1) to organise, we obtain: In this way, Z can be referred to as the value of the original mechanical properties, taking into account the variability of the CPT.
We let X and Y have a joint probability density function   , f x y , then the distribution function of Z is as follows.
Further, according to 0 y  and 0 y  , there is: Using permutation, we replace x with the variable u such that: uy x  ( 6 ) Then, ydu dx  ( 7 ) Substituting Equation (7) into Equation ( 5), there are: The collation leads to: The probability density function Clearly, X and Y are independent of each other, and we let the probability density functions be   X f x and   Y f y , respectively, then: On the other hand, the probability density function Experience has shown that CPT T and normalized mechanical properties of composites X usually follow the normal distribution.Obviously, the dimensionless CPT Y also follows a normal distribution.We let: , , Swapping the order of integration yields: We integrate the latter term of Equation ( 16), and have according to 0 y  and 0 y  , Cy zd y Cy zd y which: where  denotes the cumulative probability function of the standard normal distribution.On the other hand, the probability density function is:

Numerical simulation and analysis
Numerical simulations are carried out using the Monte Carlo method to verify the correctness of the derived equations.Then, the effect of CPT variability on the permitted values of mechanical properties of composites is illustrated by comparing the cumulative probability curves and cumulative probability density curves.The simulation parameters are set considering the actual representation of engineering.For normalized mechanical properties, the mean is 1 1000

 
, and the standard deviation is 1 100   ; for normalised CPT, the mean is 2 1   , and the standard deviation is 2 0.1   .

Monte Carlo simulation verification
The Monte Carlo simulation process is as follows: Step 1: 10,000 normalized mechanical properties are randomly generated and obey a normal distribution with mean and standard deviation Step 2: 1 normalised CPT j y is randomly generated and obeys a normal distribution with mean and standard deviation 2 0.1   .
Step 3: We calculate to obtain mechanical property values for 10,000 normalised CPT j y .
Steps 1 to 3 were repeated 10,000 times to generate 100 million mechanical property simulations.Substituting the normal distribution parameters used in the simulation into Equations ( 17) to (19), the cumulative distribution curve and the probability density curve of the mechanical properties considering the variability of the CPT are obtained by numerical integration.A comparison of the above two curves with the histogram plotted from the simulation data is shown in Figure 1 and Figure 2.  It can be seen from Figure 1 and Figure 2 that the simulation curves are in good agreement with the derived distribution formulae.For the cumulative probability curves, the mean square error between the Monte Carlo method and the method in this paper verifies the accuracy of the derived distribution formulae.Meanwhile, observing Figure 2, it can be found that the mechanical properties considering the CPT variability are skewed distributions.

Comparison of mechanical property distribution
Further, the mechanical properties considering the variability of the CPT are compared with the normalized mechanical properties, as shown in Figure 3 and Figure 4.The comparison in Figure 4 further confirms that the mechanical properties considering the CPT variability obey a skewed distribution.As can be seen from Figure 3, the mechanical properties, taking into account the CPT variability, are significantly lower than the normalized ones in the region with lower cumulative probability, which is particularly evident at the 10th percentile corresponding to the B-baseline value.

Conclusion
In this paper, the cumulative distribution function and probability density function of mechanical properties of composite materials considering the variability of CPT are derived with the precondition that both CPT and normalized mechanical properties obey normal distribution.The simulation is carried out with parameters representative of engineering practice, and the following conclusions are obtained: 1) The Monte Carlo simulation results verify the correctness of the formulas for the distribution function and probability density function of the mechanical properties of composites considering the variability of CPT.
2) The simulation compares the mechanical property distributions of composites after normalization with those considering the CPT variability.The results show that the mechanical properties are skewed after considering the CPT variability, and the permitted value of the composite B-baseline mechanical properties will be reduced.

Figure 1 .
Figure 1.Cumulative probability of mechanical properties considering CPT variability.

Figure 2 .
Figure 2. Probability density of mechanical properties considering CPT variability.