Data-driven modeling for thermo-elastic properties of vacancy-defective graphene reinforced nanocomposites with its application to functionally graded beams

Abstract

The presence of unavoidable defects in the form of atom vacancies in graphene sheets considerably deteriorates the thermo-elastic properties of graphene-reinforced nanocomposites. Since none of the existing micromechanics models is capable of capturing the effect of vacancy defect, accurate prediction of the mechanical properties of these nanocomposites poses a great challenge. Based on molecular dynamics (MD) databases and genetic programming (GP) algorithm, this paper addresses this key issue by developing a data-driven modeling approach which is then used to modify the existing Halpin–Tsai model and rule of mixtures by taking vacancy defects into account. The data-driven micromechanics models can provide accurate and efficient predictions of thermo-elastic properties of defective graphene-reinforced Cu nanocomposites at various temperatures with high coefficients of determination (R² > 0.9). Furthermore, these well-trained data-driven micromechanics models are employed in the thermal buckling, elastic buckling, free vibration, and static bending analyses of functionally graded defective graphene reinforced composite beams, followed by a detailed parametric study with a particular focus on the effects of defect percentage, content, and distribution pattern of graphene as well as temperature on the structural behaviors.

1 Introduction

Graphene exhibits superior mechanical properties, such as ultra-high elastic modulus and unsurpassed strength [1]. A great deal of effort has been made to develop lightweight and high-strength composite materials reinforced by graphene [2, 3]. For example, Kim et al. [4] reported a metal-graphene nanolayered composite whose mechanical properties are significantly improved confirmed by their transmission electron microscopy compression tests and molecular dynamics (MD) simulations. Nevertheless, most of the experimental or theoretical studies (MD simulations or continuum modelling) exclude the effect of graphene defects on the strengthening effect of composites. Structural defects that are inevitable during its growth or processing can be used to tune the local properties of graphene and thus achieve new functionalities [5]. Several atomic force microscopy nanoindentation tests demonstrated that the presence of vacancy defects significantly affects the mechanical properties of graphene [6, 7]. It was experimentally and numerically indicated that the graphene defects can effectively tailor the mechanical properties of graphene reinforced composites because the interfacial bonding between graphene and matrix is improved [8,9,10].

The thermo-elastic properties of graphene-reinforced composites are often predicted using Halpin–Tsai model and rule of mixture [11, 12]. Halpin–Tsai model is a widely used micromechanics model to estimate Young’s modulus of composites which has been employed successfully in a variety of experimental and simulated composite systems [13, 14]. On the other hand, the rule of mixture is one of the simplest theoretical models that has been developed to describe the material parameters including coefficient of thermal expansion (CTE), Poisson’s ratio, and density of graphene reinforced nanocomposites [12]. However, these theoretical models are too simple to reflect accurate results because they only consider the influencing factors of graphene content and size but ignore other key factors such as graphene defect and temperature, limiting their wider engineering applications. Thus, it is of great significance to modify the existing micromechanics models.

Recently, machine learning (ML) techniques have risen to prominence in the field of material science [15, 16]. Large databases with various features are trained using ML algorithms to obtain the models with a strong generalization ability and high accuracy when assessing the material properties of composites. For instance, the artificial neural network and support vector machine are important ML algorithms for obtaining data-driven models to predict the material properties of composites, which show superior predictive capacity [17]. However, these models are regarded as “black box” models since it is difficult for them to generate an explicit mathematical expression between the input variables and output targets. By contrast, genetic programming (GP) can derive the relationship between the inputs and outputs with explicit formulas [18]. This algorithm has been successfully used for the material properties prediction and structural behavior modeling of engineering systems [19, 20]. Hence, the GP method will be adopted to train massive datasets to obtain a data-driven Halpin–Tsai model and rule of mixture for thermo-elastic properties predictions of defective graphene reinforced nanocomposites.

Graphene-reinforced composites have been designed to be functionally graded graphene-reinforced composite structures with great success to improve structural performances [12, 21]. The thermal buckling [22, 23], elastic buckling [24,25,26], free vibration [26,27,28], and static bending properties [29] of structural components achieve remarkable enhancements when graphene is distributed in a gradient along the thickness direction. These theoretically structural behaviors are superior to those of actual cases since the graphene filler used in these structures is defect-free graphene platelet. The material parameters of nanocomposites are predicted utilizing the original Halpin–Tsai model that excludes the effect of graphene defect and temperature. To the authors’ best knowledge, the mechanical responses of functionally graded vacancy-defective graphene reinforced composite structures have not been investigated.

To fill this research gap, we develop data-driven models for predicting thermo-elastic properties of vacancy-defective graphene reinforced copper nanocomposites. First of all, the datasets including thermo-elastic properties and corresponding features of composites are collected from numerous MD simulations. Next, the datasets are trained through the GP algorithm to obtain explicit expressions to modify the existing micromechanics models. The data-driven micromechanics models are also validated with available experimental and simulated results. Finally, the models are used to predict the material parameters for analyzing the structural behaviors of functionally graded defective graphene reinforced composite beams.

2 Methods

2.1 Molecular dynamics simulation

Databases on the thermo-elastic properties of defective graphene-reinforced copper nanocomposites are generated by extensive MD simulations. The simulations are performed by large-scale atomic/molecular massively parallel simulator (LAMMPS) code [30]. The embedded-atom method (EAM) potential [31], adaptive intermolecular reactive empirical bond order (AIREBO) potential [32], and Lennard–Jones (LJ) potential are employed to describe the bonding interactions among Cu atoms, covalent interactions of C–C bonds, and van der Waal (vdW) interactions between graphene and Cu matrix, respectively. The potential parameters in detail used in the simulations refer to Refs. [3, 33, 34]

The vacancy-defective graphene is created by deleting carbon atoms randomly from a pristine graphene sheet whose size is 83.76 Å × 82.21 Å (2720 atoms). We designed six vacancy defective graphene sheets with defect percentage (D_Gr) of 0.0%, 0.2%, 0.4%, 0.6%, 0.8%, and 1.0%. Then the defective graphene is incorporated into the Cu matrix to form nanocomposites, as shown in Fig. 1. The boundary conditions of the models are periodic in three directions. The length and width of the Cu matrix are kept constant as 21a₀ (a₀ = 3.614 Å) while the height is changed to tune graphene content. By varying its height from 40a₀ to 8a₀, we design seven defective graphene/Cu nanocomposites with graphene weight fraction (W_Gr) of 0.62 wt%, 1.09 wt%, 1.41 wt%, 1.83 wt%, 2.37 wt%, 2.94 wt%, and 3.35 wt% [34]. The volume fraction (V_Gr) of graphene can be obtained accordingly by the following formula,

$$V_{{{\text{Gr}}}} = \frac{{\rho_{{{\text{Cu}}}} W_{{{\text{Gr}}}} }}{{\rho_{{{\text{Cu}}}} W_{{{\text{Gr}}}} + \rho_{{{\text{Gr}}}} \left( {1 - W_{{{\text{Gr}}}} } \right)}},$$

(1)

Fig. 1

Atomic configurations of defective graphene and its Cu nanocomposite. Tensile stress–strain curves of defective graphene reinforced nanocomposites with the effects of defect, graphene content, and temperature

where ρ_Cu and ρ_Gr are densities of pure Cu and graphene, respectively.

Afterwards, the system energy is minimized using the conjugate gradient algorithm. Then further relaxations of systems under NVT ensemble and NPT ensemble are implemented at zero pressure and specific temperature for 100 ps with a timestep of 0.5 fs, respectively. After that, uniaxial strain-controlled tensile loading is applied to the models along y-direction with a strain rate of 0.001 ps⁻¹ under the same NPT ensemble at desired temperature (T) conditions (i.e., 100 K, 200 K, … 800 K). Young’s modulus, Poisson’s ratio, and density can be extracted from the simulations. Moreover, the heating simulation is performed under an NPT ensemble with a heating rate of 1 K/ps to obtain the CTE of defective graphene-reinforced nanocomposites [35]. The atomic configurations of defective graphene and its Cu nanocomposites are visualized using OVITO software [36].

2.2 Data-driven modeling

The flowchart of data-driven modeling for predicting thermo-elastic properties of defective graphene reinforced nanocomposites is illustrated in Fig. 2. The first step is data generation based on MD simulations. Three key features (D_Gr, V_Gr, and T) are employed to build defective graphene/Cu nanocomposites. Accordingly, 384 (6 × 8 × 8) tensile simulations of composites including six groups of D_Gr, eight groups of V_Gr, and eight groups of T are implemented to determine their Young’s modulus, Poisson’s ratio, and density. In addition, 48 (6 × 8) heating simulations are executed to calculate the CTE of each simulation at eleven different temperatures. Consequently, massive data classified by three different input features and four output properties are collected, which are given in Supplementary Information.

Fig. 2

Flowchart of data-driven modeling for the prediction of thermo-elastic properties of defective graphene reinforced nanocomposites

The second step is data-driven model training via the GP algorithm (Fig. 2). The algorithm can solve symbolic regression problems through gene tree structures which can be transferred directly into symbolic mathematical formulas [18, 37]. A free machine learning platform “Genetic Programming Toolbox for the Identification of Physical Systems (GPTIPS)” is used for multigene symbolic regression training [37]. Table 1 summarizes the parameters used in the algorithm. The population size and generation number are set to be 100 which controls the computational efficiency and accuracy of data-driven models. The functions and terminals in the tree-like structure are randomly selected from the sets of functions and terminals. At first, an initial random population of tree-structured symbolic expressions is generated with the maximum tree number and depth of 5 and 3, respectively. The larger the parameters are, the more complicated the evolved data-driven models will be. Next, the GP algorithm evaluates its fitness. If it does not satisfy the termination criteria, it will choose an individual for reproduction, crossover, and mutation to generate a new generation after iterations. The crossover and mutation probabilities are 0.85 and 0.1, respectively. The process is iterated until the termination criteria are satisfied. As a result, we can obtain a multigene data-driven model as

$$Y^{{{\text{ML}}}} = b_{0} + b_{1} G_{1} + b_{2} G_{2} + \cdots + b_{m} G_{m} ,$$

(2)

where G_m represents the gene tree expression, m denotes the number of genes, b₀ is the bias coefficient, and b₁, b₂, …, b_m are weight coefficients to scale each gene tree. The accuracy of the data-driven micromechanics models is assessed using root mean square error (RMSE) and coefficient of determination (R²) as

$${\text{RMSE}} = \sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {\left( {Y_{i}^{{\text{MD}}} - Y_{i}^{{\text{ML}}} } \right)^{2} } }}{n}} ,$$

(3a)

$$R^{2} = \frac{{\sum\nolimits_{i = 1}^{n} {\left( {Y_{i}^{{\text{MD}}} - \overline{Y}^{{\text{MD}}} } \right)\left( {Y_{i}^{{\text{ML}}} - \overline{Y}^{{\text{ML}}} } \right)} }}{{\sqrt {\sum\nolimits_{i = 1}^{n} {\left( {Y_{i}^{{\text{MD}}} - \overline{Y}^{{\text{MD}}} } \right)^{2} } } \sqrt {\sum\nolimits_{i = 1}^{n} {\left( {Y_{i}^{{\text{ML}}} - \overline{Y}^{{\text{ML}}} } \right)^{2} } } }}$$

(3b)

in which ${Y}_{i}^{\mathrm{MD}}$ and ${Y}_{i}^{\mathrm{ML}}$ are the thermo-elastic properties of the ith composite obtained by MD simulations and ML predictions, respectively; $\overline{Y }$ represents the average values of outputs and n denotes the total number of samples.

Table 1 Parameters used in the genetic programming

Other ML algorithms including k-nearest neighbor (KNN), decision tree (DT), and random forest (RF) are further used to train the MD datasets to examine their performances for the comparison with GP algorithm. Scikit-learn is adopted to perform ML training [38].

The third step is thermo-elastic properties prediction of defective graphene reinforced nanocomposites with new features using the well-trained data-driven micromechanics models. In the last step, the predicted material parameters are used for structural analysis (e.g., thermal buckling, elastic buckling, free vibration, and static bending) of functionally graded defective graphene reinforced composite beams considering the effects of graphene defect, graphene content, and temperature.

3 Results and discussion

3.1 Thermo-elastic properties of defective graphene reinforced nanocomposites

After tensile simulations of the composites, we can obtain their stress–strain curves with various graphene defect percentages, graphene contents, and temperatures, as plotted in Fig. 1. The Young’s moduli are extracted from the initial slope of these curves. Figure 3 shows the thermo-elastic properties of defective graphene/Cu nanocomposites at different conditions. The effect of graphene defect (between 0 and 1%) on the material properties is presented in Fig. 3A. The Young’s modulus decreases gradually from ~ 155.44 GPa at the defect percentage of 0% (perfect graphene) to ~ 149.89 GPa at the defect percentage of 1%, reduced by ~ 3.57%. The reason is that the presence of vacancy defect in graphene deteriorates its stiffness [7], leading to the decreased mechanical properties of its Cu composites. In contrast, the CTE, Poisson’s ratio, and density are insensitive to the variation of graphene defects due to the small defect degree in graphene.

Fig. 3

Young’s modulus, CTE, Poisson’s ratio, and density of defective graphene reinforced nanocomposites with different A defect percentages (14.47 vol% and 300 K), B graphene contents (300 K and 1%), and C temperatures (1% and 14.47 vol%)

With regards to the effect of graphene content, we can see from Fig. 3B that Young’s modulus is 149.89 GPa when the graphene content reaches 14.47 vol%, 86.04% larger than that at graphene content of 2.95 vol%. Furthermore, there is a remarkable drop in CTE from ~ 14.68 × 10^–6 K⁻¹ at graphene content of 2.95 vol% to ~ 9.89 × 10^–6 K⁻¹ at graphene content of 14.47 vol%, corresponding to a reduction of ~ 32.63%. This is because graphene has negative CTE that makes its composites achieve a lower CTE at a larger content [35]. A continuous decrease in Poisson’s ratio can also be seen as the graphene content increases. More importantly, more graphene content leads to a lower density of composites since the graphene is far lighter than Cu [39, 40], which enables us to design lightweight yet strong composite materials.

Temperature is also an important influencing factor on thermo-elastic properties of defective graphene reinforced nanocomposites, as displayed in Fig. 3C. The Young’s modulus reduces by ~ 20.94% to ~ 124.32 GPa at 800 K compared to that at 100 K. Higher temperature also results in smaller density due to the volume expansion. However, the temperature has positive effects on the CTE and Poisson’s ratio of composites. This is due to the increased thermal vibration of atoms caused by the high temperature [34, 35].

For the prediction of thermo-elastic properties of graphene reinforced composites, there have been many micromechanics models such as Halpin–Tsai model and the rule of mixture that are given by [12],

$$E_{{\text{c}}} = \frac{{1 + \xi \eta V_{{{\text{Gr}}}} }}{{1 - \eta V_{{{\text{Gr}}}} }}E_{{{\text{Cu}}}} ,$$

(4a)

$$\alpha_{\text{c}} = \alpha_{{\text{Gr}}} V_{{\text{Gr}}} + \alpha_{{\text{Cu}}} V_{{\text{Cu}}} ,$$

(4b)

$$\nu_{\text{c}} = \nu_{{\text{Gr}}} V_{{\text{Gr}}} + \nu_{{\text{Cu}}} V_{{\text{Cu}}} ,$$

(4c)

$$\rho_{\text{c}} = \rho_{{\text{Gr}}} V_{{\text{Gr}}} + \rho_{{\text{Cu}}} V_{{\text{Cu}}} ,$$

(4d)

where the size parameter $\xi = 2\left( {{{l_{{\text{Gr}}} } \mathord{\left/ {\vphantom {{l_{{\text{Gr}}} } {t_{{\text{Gr}}} }}} \right. \kern-\nulldelimiterspace} {t_{{\text{Gr}}} }}} \right)$; l_Gr and t_Gr are length and thickness of graphene, respectively; the material coefficient $\eta = \frac{{\left( {{{E_{{\text{Gr}}} } \mathord{\left/ {\vphantom {{E_{{\text{Gr}}} } {E_{{\text{Cu}}} }}} \right. \kern-\nulldelimiterspace} {E_{{\text{Cu}}} }}} \right) - 1}}{{\left( {{{E_{{\text{Gr}}} } \mathord{\left/ {\vphantom {{E_{{\text{Gr}}} } {E_{{\text{Cu}}} }}} \right. \kern-\nulldelimiterspace} {E_{{\text{Cu}}} }}} \right) + \xi }}$. Although these models have been successfully applied in many experimental and theoretical systems [11, 12, 34], they only include the influencing factors of graphene content and size but exclude other key factors including graphene defect and temperature. Therefore, we aim to modify the micromechanics models by adding modification functions f_E,α,ν,ρ(D_Gr, V_Gr, T) obtained through the GP algorithm in the next section:

$$E_{{\text{c}}} = \frac{{1 + \xi \eta V_{{\text{Gr}}} }}{{1 - \eta V_{{\text{Gr}}} }}E_{{\text{Cu}}} \times f_{\text{E}} (D_{{\text{Gr}}} ,V_{{\text{Gr}}} ,T),$$

(5a)

$$\alpha_{\text{c}} = \left( {\alpha_{{\text{Gr}}} V_{{\text{Gr}}} + \alpha_{{\text{Cu}}} V_{{\text{Cu}}} } \right) \times f_{\alpha } (D_{{\text{Gr}}} ,V_{{\text{Gr}}} ,T),$$

(5b)

$$\nu_{\text{c}} = \left( {\nu_{{\text{Gr}}} V_{{\text{Gr}}} + \nu_{{\text{Cu}}} V_{{\text{Cu}}} } \right) \times f_{\nu } (D_{{\text{Gr}}} ,V_{{\text{Gr}}} ,T),$$

(5c)

$$\rho_{\text{c}} = \left( {\rho_{{\text{Gr}}} V_{{\text{Gr}}} + \rho_{{\text{Cu}}} V_{{\text{Cu}}} } \right) \times f_{\rho } (D_{{\text{Gr}}} ,V_{{\text{Gr}}} ,T).$$

(5d)

3.2 Data-driven modeling based on MD simulation and GP algorithm

3.2.1 Data-driven model training

The MD datasets are well trained using the GP algorithm where the proportions of the training set, validation set, and test set are 60%, 20%, and 20%, respectively. Then we can obtain the gene tree-like structures of data-driven modification functions to modify micromechanics models for thermo-elastic properties prediction of defective graphene/Cu nanocomposites, as shown in Fig. 4. The corresponding mathematical formulas can be derived by decoding these tree-like genes. Substituting these gene expressions as well as the bias and weighting coefficients of each gene shown in Table 2 into Eq. 2, one can readily obtain formulations of modification functions of Halpin–Tsai model for Young’s modulus and rule of mixture models for CTE, Poisson’s ratio, and density. The specific modification functions for micromechanics models are given in Eq. 6. Since defect percentage is too small to have obvious influences on CTE, Poisson’s ratio, and density of the composite, this parameter is not included in the data-driven models.

$$f_{\text{E}} (D_{\text{Gr}} ,V_{\text{Gr}} ,T) = 1.1 - 0.92V_{\text{Gr}} - 0.118\left( {\frac{T}{{T_{0} }}} \right) + 2.22V_{\text{Gr}}^{2} - 31.1D_{\text{Gr}} V_{\text{Gr}} + 0.112V_{\text{Gr}}^{2} \left( {\frac{T}{{T_{0} }}} \right) + 0.112V_{\text{Gr}} \left( {\frac{T}{{T_{0} }}} \right)^{2} ,$$

(6a)

$$f_{\alpha } (V_{\text{Gr}} ,T) = 0.93 - 4.4V_{\text{Gr}} + 0.066\left( {\frac{T}{{T_{0} }}} \right) + 3.77V_{\text{Gr}} \left( {\frac{T}{{T_{0} }}} \right) - 0.837V_{\text{Gr}} \left( {\frac{T}{{T_{0} }}} \right)^{2} ,$$

(6b)

$$f_{\nu } (V_{\text{Gr}} ,T) = 1.47 - 6.85V_{\text{Gr}} + 0.15\left( {\frac{T}{{T_{0} }}} \right) + 7.27V_{\text{Gr}}^{2} + 0.57V_{\text{Gr}} \left( {\frac{T}{{T_{0} }}} \right),$$

(6c)

$$f_{\rho } (V_{\text{Gr}} ,T) = 1.02 - 0.591V_{\text{Gr}}^{2} - 0.0152\left( {\frac{T}{{T_{0} }}} \right),$$

(6d)

where T₀ = 300 K is the reference temperature.

Fig. 4

Gene tree structures of modification functions for A Young’s modulus, B CTE, C Poisson’s ratio, and D density

Table 2 Bias and weighting coefficients of each gene for modification functions of Young’s modulus (E), CTE (α), Poisson’s ratio (ν), and density (ρ)

3.2.2 Performance evaluation of data-driven models

The data-driven model predictions and MD simulation results on thermo-elastic properties of defective graphene reinforced composites are graphically compared in Fig. 5A–D. Fig. 6 further presents the comparisons between simulations and predictions with regard to the training dataset, validation dataset, and testing dataset. The data-driven models can predict thermo-elastic properties of the composites with high accuracy and low errors in all datasets. For Young’s modulus predicted by the data-driven Halpin–Tsai model, the RMSE are 1.99%, 1.75%, and 1.88% regarding the training set, validation set, and test set, respectively. By contrast, the corresponding R² are 0.9311, 0.9385, and 0.9471, respectively (Fig. 5E). It is concluded that the developed data-driven Halpin–Tsai model accurately captures the effects of input variables including defect percentage, graphene content, and temperature on Young’s modulus of the composites. Apart from Young’s modulus, other thermo-elastic properties can be predicted using the data-driven rule of mixture very well with RMSE lower than 3% and R² higher than 0.9. These evaluation indices confirm the accuracy and effectiveness of the developed data-driven micromechanics models.

Fig. 5

Thermo-elastic properties predictions using data-driven micromechanics models. Comparisons between the Predictions and simulations on A Young’s modulus, B CTE, C Poisson’s ratio, and D density of the composites. E The coefficients of determination for thermo-elastic properties datasets. F High efficiency of data-driven models in comparison with that of MD simulations

Fig. 6

Comparison between the data-driven model predictions and MD simulations on A Young’s modulus, B CTE, C Poisson’s ratio, and D density

Another advantage of the data-driven model is high efficiency, as illustrated in Fig. 5F. The computation cost is evaluated using the central processing unit (CPU) time used for each MD simulation. The tensile simulation of the composite is carried out using LAMMPS software on a 24-core supercomputer (Gadi, NCI Australia) that spends about 20 CPU hours on average, equivalent to approximately 480 CPU hours on a single-core CPU. In comparison, only ~ 1 min is taken for the prediction using data-driven models. Such a remarkable reduction of computation cost for the data-driven model effectively overcome the disadvantage of the low efficiency of MD simulations. It is easy for users to employ the data-driven micromechanics models for thermo-elastic properties evaluation of defective graphene reinforced nanocomposites and structural analysis of their composite structures.

Moreover, we also examined the performances of other ML methods such as KNN, DT, and RF for the estimation of material properties of defective graphene reinforced Cu nanocomposites, together with that of the GP algorithm, in Fig. S1. In addition to the KNN algorithm, all the others are capable of accurately estimating the material properties with high R² values. But unlike the GP algorithm, these ML algorithms cannot generate the explicit mathematical expression between the inputs and outputs. Hence, the GP algorithm is selected to develop the micromechanical models in this paper.

3.2.3 Validation of data-driven models

Furthermore, we verify the data-driven micromechanics model with available experimental and simulated data to indicate its generalization (Fig. 7). Fig. 7A shows the comparisons on Young’s modulus of graphene/Cu composites between the data-driven model predictions and experimental as well as simulated results from Refs. [14, 41,42,43] in which all values are distributed around the diagonal. It is clear from Fig. 7B that all R² values are more than 0.9, demonstrating that the data-driven model can accurately predict the experimental results and other MD simulation results.

Fig. 7

Validation of data-driven micromechanics model. A Comparison between the data-driven model predicted Young’s modulus and the experimental results by Hwang et al. [41], Xiong et al. [42], Cao et al. [43], as well as simulation results by Zhang et al. [14] B the coefficients of determination for Young’s modulus

3.3 Mechanical behaviors of functionally graded defective graphene reinforced composite beams

3.3.1 Theoretical formulations

The well-trained data-driven micromechanics models (Eqs. 5 and 6) can be used to accurately predict thermo-elastic properties of the defective graphene/Cu composite for mechanical analysis of its composite beam structure. Shown in Fig. 8 is a slender multilayer (N_L) defective graphene reinforced composite beam with thickness h, length L and width b where defective graphene is uniformly distributed within each layer with the graphene content changing from layer to layer along the thickness direction. Each layer is assumed to be isotropic homogeneous with the same thickness h/N_L. Three distribution patterns of defective graphene across the beam thickness are designed (Fig. 8) where darker color denotes a higher graphene content. Pattern U-W_Gr is a special case that is an isotropic homogeneous beam with the same graphene content in each layer. Patterns X-W_Gr and O-W_Gr are symmetrical distributions in which graphene content is the maximum in the surface layers for the X-W_Gr beam while this is inversed for the O-W_Gr beam.

Fig. 8

A multilayer defective graphene reinforced composite beam and its three different defective graphene distribution patterns along the thickness direction

The volume fractions V_Gr(k) of the kth layer for the three distributions are governed by the following expressions [24],

U-W_Gr: $V_{\text{Gr}} (k) = V_{\text{Gr}}$(7a)

X-W_Gr: $V_{\text{Gr}} (k) = 2V_{\text{Gr}} \left| {2k - N_{L} - 1} \right|/N_{L}$(7b)

O-W_Gr: $V_{\text{Gr}} (k) = 2V_{\text{Gr}} \left( {1 - \left| {2k - N_{L} - 1} \right|/N_{L} } \right)$(7c)

where k = 1, 2, …, N_L; N_L denotes the total number of layers of the functionally graded multilayer composite beam. The total volume fraction of graphene V_Gr is determined by Eq. 1.

The beam is described based on Euler–Bernoulli beam theory where the displacement field in the X- and Z-axis of an arbitrary point takes the form of

$$\overline{U} \left( {X,Z,T_{t} } \right) = U\left( {X,T_{t} } \right) - Z\frac{{\partial W\left( {X,T_{t} } \right)}}{\partial X},$$

(8a)

$$\overline{W} \left( {X,Z,T_{t} } \right) = W\left( {X,T_{t} } \right)$$

(8b)

in which U(X,T_t) and W(X,T_t) represent the displacement components in the midplane; T_t is time. The axial force N and bending moment M are calculated by

$$N = A_{11} \frac{\partial U}{{\partial X}} - B_{11} \frac{{\partial^{2} W}}{{\partial X^{2} }} - N^{T} ,$$

(9a)

$$M = B_{11} \frac{\partial U}{{\partial X}} - D_{11} \frac{{\partial^{2} W}}{{\partial X^{2} }} - M^{T} .$$

(9b)

The stiffness components, as well as axial force and bending moment caused by thermal stresses are defined as

$$\left\{ {A_{11} ,B_{11} ,D_{11} } \right\} = \int_{ - h/2}^{h/2} {\frac{{E_{\text{c}} (Z)}}{{1 - \nu_{\text{c}}^{2} (Z)}}} \left\{ {1,Z,Z^{2} } \right\}{\varvec{d}}Z = \sum\limits_{k = 1}^{{N_{L} }} {\int_{{Z_{k} }}^{{Z_{k + 1} }} {\frac{{E_{\text{c}} (k)}}{{1 - \nu_{\text{c}}^{2} (k)}}} \left\{ {1,Z,Z^{2} } \right\}{\varvec{d}}Z} ,$$

(10a)

$$\left\{ {N^{T} ,M^{T} } \right\} = \int_{ - h/2}^{h/2} {\frac{{E_{\text{c}} (Z)}}{{1 - \nu_{\text{c}}^{2} (Z)}}} \alpha_{\text{c}} (Z)\Delta T\left\{ {1,Z} \right\}{\varvec{d}}Z = \sum\limits_{k = 1}^{{N_{L} }} {\int_{{Z_{k} }}^{{Z_{k + 1} }} {\frac{{E_{\text{c}} (k)}}{{1 - \nu_{\text{c}}^{2} (k)}}} \alpha_{\text{c}} (k)\Delta T\left\{ {1,Z} \right\}{\varvec{d}}Z} .$$

(10b)

The equations of motion for the functionally graded defective graphene reinforced composite beam with the axial inertia term being neglected and the effects of thermal load N^T, axial compressive force P, and transverse uniformly distributed load Q being included are derived as follows,

$$A_{11} \frac{{\partial^{2} U}}{{\partial X^{2} }} - B_{11} \frac{{\partial^{3} W}}{{\partial X^{3} }} = 0,$$

(11a)

$$\left( {D_{11} - \frac{{B_{11}^{2} }}{{A_{11} }}} \right)\frac{{\partial^{4} W}}{{\partial X^{4} }} + N^{T} \frac{{\partial^{2} W}}{{\partial X^{2} }} + P\frac{{\partial^{2} W}}{{\partial X^{2} }} + I_{1} \frac{{\partial^{2} W}}{{\partial T_{t}^{2} }} = Q,$$

(11b)

where the inertial component is defined as

$$I_{1} = \int_{ - h/2}^{h/2} {\rho_{\text{c}} (Z)} {\varvec{d}}Z = \sum\limits_{k = 1}^{{N_{L} }} {\int_{{Z_{k} }}^{{Z_{k + 1} }} {\rho_{\text{c}} (k)} {\varvec{d}}Z} .$$

(12)

A clamped–clamped (C–C) beam is considered in the present study. The associated boundary conditions satisfy

$$U\user2{ = }W = \frac{{{\varvec{d}}W}}{{{\varvec{d}}X}} = 0\,\text{at }X = 0,L,$$

(13)

With the introduction of the following dimensionless quantities:

$$\begin{gathered} x = \frac{X}{L},\left\{ {u,w} \right\} = \frac{{\left\{ {U,W} \right\}}}{h},\beta = \frac{h}{L},\left\{ {a_{11} ,b_{11} ,d_{11} ,} \right\} = \left\{ {\frac{{A_{11} }}{{A_{110} }},\frac{{B_{11} }}{{A_{110} h}},\frac{{D_{11} }}{{A_{110} h^{2} }}} \right\}, \hfill \\ i_{1} = \frac{{I_{1} }}{{I_{10} }},\left\{ {n^{T} ,m^{T} } \right\} = \left\{ {\frac{{N^{T} }}{{A_{110} }},\frac{{M^{T} }}{{A_{110} h}}} \right\},\left\{ {p,q} \right\} = \left\{ {\frac{P}{{A_{110} }},\frac{QL}{{A_{110} \beta }}} \right\},t = \frac{{T_{t} }}{L}\sqrt {\frac{{A_{110} }}{{I_{10} }}} . \hfill \\ \end{gathered}$$

(14)

where A₁₁₀ and I₁₀ are the values of A₁₁ and I₁ of an isotropic homogeneous beam made of pure Cu at room temperature. The equations of motion in a dimensionless form are rewritten as

$$a_{11} \frac{{\partial^{2} u}}{{\partial x^{2} }} - b_{11} \beta \frac{{\partial^{3} w}}{{\partial x^{3} }} = 0,$$

(15a)

$$\left( {d_{11} - \frac{{b_{11}^{2} }}{{a_{11} }}} \right)\beta^{2} \frac{{\partial^{4} w}}{{\partial x^{4} }} + n^{T} \frac{{\partial^{2} w}}{{\partial x^{2} }} + p\frac{{\partial^{2} w}}{{\partial x^{2} }} + i_{1} \frac{{\partial^{2} w}}{{\partial t^{2} }} = q.$$

(15b)

Meanwhile, the boundary condition shown in Eq. 13 can be transformed into a dimensionless form as

$$u\user2{ = }w = \frac{{{\varvec{d}}w}}{{{\varvec{d}}x}} = 0\,\text{at }x = 0,1.$$

(16)

• (1) Thermal buckling analysis

Neglecting the effects of axial compressive force, inertia term, and uniform load, the equations of motion (Eq. 15) are simplified as the following equations of equilibrium for thermal buckling analysis of the composite beam

$$a_{11} \frac{{\partial^{2} u}}{{\partial x^{2} }} - b_{11} \beta \frac{{\partial^{3} w}}{{\partial x^{3} }} = 0,$$

(17a)

$$\left( {d_{11} - \frac{{b_{11}^{2} }}{{a_{11} }}} \right)\beta^{2} \frac{{\partial^{4} w}}{{\partial x^{4} }} + n^{T} \frac{{\partial^{2} w}}{{\partial x^{2} }} = 0.$$

(17b)

The general solutions of Eq. 17 can be readily obtained as

$$w\left( x \right) = C_{1} \sin (\theta x) + C_{2} \cos (\theta x) + C_{3} x + C_{4} ,$$

(18a)

$$u\left( x \right) = \frac{{b_{11} \beta }}{{a_{11} }}\left[ {\theta C_{1} \cos (\theta x) - \theta C_{2} \sin (\theta x)} \right] + C_{5} x + C_{6} ,$$

(18b)

where $\theta^{2} = \frac{{n^{T} }}{{\left( {d_{11} - {{b_{11}^{2} } \mathord{\left/ {\vphantom {{b_{11}^{2} } {a_{11} }}} \right. \kern-\nulldelimiterspace} {a_{11} }}} \right)\beta^{2} }}$; C₁, C₂, …, C₆ are unknown constants to be determined from the boundary conditions. Substituting the displacement solutions into the boundary conditions yields the following matrix equation that is composed of 6 homogeneous algebraic equations

$$\left[ {{\text{H}}(\theta )} \right]\left\{ {\text{C}} \right\} = \left\{ {0} \right\},$$

(19)

This equation has a non-trivial solution when its determinant is equal to zero

$$\det \left[ {{\text{H}}(\theta )} \right] = 0.$$

(20)

The matrix H(θ) is given in Eq. 33 in Appendix.

• (2) Elastic buckling analysis

Suppose the beam is subjected to axial compressive force P, the equations of equilibrium for elastic buckling analysis of the beam are

$$a_{11} \frac{{\partial^{2} u}}{{\partial x^{2} }} - b_{11} \beta \frac{{\partial^{3} w}}{{\partial x^{3} }} = 0,$$

(21a)

$$\left( {d_{11} - \frac{{b_{11}^{2} }}{{a_{11} }}} \right)\beta^{2} \frac{{\partial^{4} w}}{{\partial x^{4} }} + p\frac{{\partial^{2} w}}{{\partial x^{2} }} = 0.$$

(21b)

The form of general solutions is the same as that for thermal buckling analysis as follows,

$$w\left( x \right) = C_{1} \sin (\gamma x) + C_{2} \cos (\gamma x) + C_{3} x + C_{4} ,$$

(22a)

$$u\left( x \right) = \frac{{b_{11} \beta }}{{a_{11} }}\left[ {\gamma C_{1} \cos (\gamma x) - \gamma C_{2} \sin (\gamma x)} \right] + C_{5} x + C_{6}$$

(22b)

in which $\gamma^{2} = \frac{p}{{\left( {d_{11} - {{b_{11}^{2} } \mathord{\left/ {\vphantom {{b_{11}^{2} } {a_{11} }}} \right. \kern-\nulldelimiterspace} {a_{11} }}} \right)\beta^{2} }}$. Likewise, inserting the displacements into boundary conditions results in a set of homogeneous equations whose determinant should be zero to guarantee a non-trivial solution

$$\left[ {{\text{H}}(\gamma )} \right]\left\{ {\text{C}} \right\} = \left\{ {0} \right\},$$

(23)

$$\det \left[ {{\text{H}}(\gamma )} \right] = 0.$$

(24)

Appendix Eq. 34 gives the detailed expression of the matrix H(γ).

• (3) Free vibration analysis

The equations of motion for the functionally graded defective graphene reinforced composite beam without the effects of thermal load and other external loads are given as

$$a_{11} \frac{{\partial^{2} u}}{{\partial x^{2} }} - b_{11} \beta \frac{{\partial^{3} w}}{{\partial x^{3} }} = 0,$$

(25a)

$$\left( {d_{11} - \frac{{b_{11}^{2} }}{{a_{11} }}} \right)\beta^{2} \frac{{\partial^{4} w}}{{\partial x^{4} }} + i_{1} \frac{{\partial^{2} w}}{{\partial t^{2} }} = 0.$$

(25b)

Expressing the dynamic displacements in harmonic vibration as

$$u = \tilde{u}e^{{{\varvec{i}}\omega t}} ,$$

(26a)

$$w = \tilde{w}e^{{{\varvec{i}}\omega t}} .$$

(26b)

in which ω is the natural frequency. Substitution of Eq. 26 into Eq. 25 leads to the following equations governing free vibration behaviors of the beam

$$a_{11} \frac{{\partial^{2} \tilde{u}}}{{\partial x^{2} }} - b_{11} \beta \frac{{\partial^{3} \tilde{w}}}{{\partial x^{3} }} = 0,$$

(27a)

$$\left( {d_{11} - \frac{{b_{11}^{2} }}{{a_{11} }}} \right)\beta^{2} \frac{{\partial^{4} \tilde{w}}}{{\partial x^{4} }} - i_{1} \omega^{2} \tilde{w} = 0.$$

(27b)

The general solutions of flexural and axial mode shape functions are

$$\tilde{w}\left( x \right) = C_{1} \sin (\lambda x) + C_{2} \cos (\lambda x) + C_{3} \sinh (\lambda x) + C_{4} \cosh (\lambda x),$$

(28a)

$$\tilde{u}\left( x \right) = \frac{{b_{11} \beta }}{{a_{11} }}\left[ {\lambda C_{1} \cos (\lambda x) - \lambda C_{2} \sin (\lambda x) + \lambda C_{3} \cosh (\lambda x) + \lambda C_{4} \sinh (\lambda x)} \right] + C_{5} x + C_{6} ,$$

(28b)

where $\lambda^{4} = \frac{{i_{1} \omega^{2} }}{{\left( {d_{11} - {{b_{11}^{2} } \mathord{\left/ {\vphantom {{b_{11}^{2} } {a_{11} }}} \right. \kern-\nulldelimiterspace} {a_{11} }}} \right)\beta^{2} }}$. Similarly, the following matrix equation is obtained where the determinant is zero to determine its non-trivial solution

$$\left[ {{\text{H}}(\lambda )} \right]\left\{ {\text{C}} \right\} = \left\{ {0} \right\},$$

(29)

$$\det \left[ {{\text{H}}(\lambda )} \right] = 0.$$

(30)

The matrix H(λ) is shown in Eq. 35 in Appendix.

• (4) Static bending analysis

For a beam under transverse uniform load, the equations of equilibrium for static bending analysis are

$$a_{11} \frac{{\partial^{2} u}}{{\partial x^{2} }} - b_{11} \beta \frac{{\partial^{3} w}}{{\partial x^{3} }} = 0,$$

(31a)

$$\left( {d_{11} - \frac{{b_{11}^{2} }}{{a_{11} }}} \right)\beta^{2} \frac{{\partial^{4} w}}{{\partial x^{4} }} = q.$$

(31b)

The general solutions of Eq. 31 are easily obtained as

$$w\left( x \right) = \frac{1}{24}\mu x^{4} + C_{1} x^{3} + C_{2} x^{2} + C_{3} x + C_{4} ,$$

(32a)

$$u\left( x \right) = \frac{{b_{11} \beta }}{{a_{11} }}\left[ {\frac{1}{6}\mu x^{3} + 3C_{1} x^{2} } \right] + C_{5} x + C_{6} ,$$

(32b)

in which $\mu = \frac{q}{{\left( {d_{11} - {{b_{11}^{2} } \mathord{\left/ {\vphantom {{b_{11}^{2} } {a_{11} }}} \right. \kern-\nulldelimiterspace} {a_{11} }}} \right)\beta^{2} }}$.

3.3.2 Validation

At first, we validate the present theoretical formulations by comparing our analytical solutions with available numerical solutions. Table 3 compares the critical buckling temperature change, dimensionless critical buckling load, dimensionless fundamental natural frequency, and dimensionless maximum deflection of C–C graphene reinforced composite beams with those from Refs. [22, 24, 26, 29] The material properties used in the examples can be found in this literature. The original Halpin–Tsai model is used to evaluate the material parameters of graphene reinforced composites. As can be observed, our analytical results are in good agreement with the existing ones, indicating the validity of the theoretical formulations.

Table 3 Comparisons of critical buckling temperature change ΔT_cr, dimensionless critical buckling load p_cr, dimensionless fundamental natural frequency ω₁, and dimensionless maximum deflection w_max of C–C graphene reinforced composite beams (X-W_Gr)

3.3.3 Parametric analysis

In what follows, the functionally graded defective graphene reinforced composite beams with a total number of layers N_L = 10 and a slenderness ratio L/h = 60 are considered. Each layer is made by vacancy-defective graphene/Cu composite whose material parameters at room temperature are [34, 35]: E_Cu = 65.79 GPa, α_Cu = 16.51 × 10^–6 K⁻¹, ν_Cu = 0.387, ρ_Cu = 8.80 g/cm³; E_Gr = 929.57 GPa, α_Gr = –3.98 × 10^–6 K⁻¹, ν_Gr = 0.220, ρ_Gr = 1.80 g/cm³, l_Gr = 83.76 Å, t_Gr = 3.4 Å. Data-driven micromechanics models shown in Eqs. 5 and 6 with these material parameters are used to analyze the effects of graphene defects, graphene contents, graphene distribution patterns, and temperatures on thermal buckling, elastic buckling, free vibration, and static bending of functionally graded defective graphene reinforced composite beams.

Fig. 9 compares the thermal buckling behaviors of functionally graded defective graphene reinforced composite beam with different graphene contents and graphene defect percentages. The results from Fig. 9A show that the beam with X-W_Gr graphene distribution is capable of carrying higher thermal loads than the beams with the other two graphene distribution patterns. The reason is that the X-W_Gr beam with more graphene distributed closely to surface layers achieves the highest beam stiffness [22, 24]. Moreover, the critical buckling temperature changes increase for all functionally graded defective graphene reinforced composite beams with increasing graphene contents. Taking the X-W_Gr beam as an example, the ΔT_cr is ~ 71.15 K at a graphene content of 1.5 wt%, ~ 29.50% larger than that of a pure Cu beam.

Fig. 9

Effects of A graphene contents and B graphene defect percentages on the critical buckling temperature change ΔT_cr of functionally graded defective graphene reinforced composite beams

The effects of graphene defect on the critical buckling temperatures are investigated in Fig. 9B for X-W_Gr defective graphene reinforced composite beams. There is a gradual decrease in ΔT_cr from ~ 71.98 K for pristine graphene reinforced composite beam to ~ 71.15 K for defective graphene reinforced composite beam with a defect percentage of 1%, corresponding to a reduction of ~ 1.15%. This is because the increasing defect in graphene leads to its performance deterioration, giving rise to the reduced mechanical properties of its composites. The detailed ΔT_cr of beams with various graphene contents and defect percentages are listed in Tables 4 and 5.

Table 4 Effects of graphene contents on the critical buckling temperature change ΔT_cr, dimensionless critical buckling load p_cr, dimensionless fundamental natural frequency ω₁, and dimensionless maximum deflection w_max of functionally graded defective graphene reinforced composite beams (C–C, L/h = 60, T = 300 K, D_Gr = 1%, X-W_Gr)

Table 5 Effects of graphene defect percentages on the critical buckling temperature change ΔT_cr, dimensionless critical buckling load p_cr, dimensionless fundamental natural frequency ω₁, and dimensionless maximum deflection w_max of functionally graded defective graphene reinforced composite beams (C–C, L/h = 60, T = 300 K, W_Gr = 1.5 wt%, X-W_Gr)

The elastic buckling responses of the composite beams with different graphene contents, defect percentages, and temperatures are plotted in Fig. 10. It is clear from Fig. 10A that the p_cr increases as the graphene content rises for all U-W_Gr, X-W_Gr, and O-W_Gr beams where the X-W_Gr beam exhibits the largest buckling resistance capability. A considerable increase of ~ 37.29% from ~ 0.001247 for pure Cu beam to ~ 0.001712 for X-W_Gr (1.5 wt%) beam can be observed. Furthermore, the dimensionless critical buckling loads drop with the increase of graphene defect percentages for various graphene contents (Fig. 10B). The decreased rate of p_cr is about 3.49% for X-W_Gr beam with graphene content of 1.5 wt% when the defect percentage increases from 0.0 to 1.0%. The temperature also has a negative effect on the critical buckling load that decreases to ~ 0.001694, reduced by 1.05%, for the beam with D_Gr of 1.0% at 400 K compared to that at 300 K, as displayed in Fig. 10C. Tables 4, 5 and 6 tabulate the representative dimensionless critical buckling loads of the beams with various influencing factors as well for direct numeric comparisons.

Fig. 10

Effects of A graphene contents, B graphene defect percentages, and C temperatures on the dimensionless critical buckling load p_cr of functionally graded defective graphene reinforced composite beams

Table 6 Effects of temperatures on the critical buckling temperature change ΔT_cr, dimensionless critical buckling load p_cr, dimensionless fundamental natural frequency ω₁, and dimensionless maximum deflection w_max of functionally graded defective graphene reinforced composite beams (C–C, L/h = 60, D_Gr = 1%, W_Gr = 1.5 wt%, X-W_Gr)

Moreover, Fig. 11 shows the free vibration characteristics of functionally graded defective graphene reinforced composite beams with different graphene contents, defect percentages, and temperatures. More graphene embedding into the metal matrix leads to a larger stiffness for the graphene/Cu composites, giving rise to an increased fundamental natural frequency of the composite beam with any graphene distribution patterns [26]. Among the three types of graphene distributions, the X-W_Gr beam has the biggest fundamental natural frequency which is ~ 0.1515 when graphene content reaches 1.5 wt%, as illustrated in Fig. 11A. On the other hand, the introduction of vacancy defect in graphene degrades the material properties of itself and its nanocomposites, resulting in the declined fundamental natural frequencies of the composite beams (Fig. 11B). Likewise, a negative effect of temperatures on the fundamental natural frequency of the composite beams can also be seen in Fig. 11C when increasing the temperature from 300 to 400 K. This is because the material properties of composites become weak when they are under high temperatures. Several representative dimensionless fundamental natural frequencies of the beams under various conditions are summarized in Tables 4, 5 and 6.

Fig. 11

Effects of A graphene contents, B graphene defect percentages, and C temperatures on the dimensionless fundamental natural frequency ω₁ of functionally graded defective graphene reinforced composite beams

Finally, the static bending behaviors of functionally graded defective graphene reinforced composite beams under transverse uniform loads with different graphene contents, defect percentages, and temperatures are presented in Fig. 12. The dimensionless maximum deflection declines with the rise of graphene contents for each composite beam since the incorporation of graphene effectively improves the stiffness of the Cu nanocomposites [29]. More importantly, a remarkable reduction of ~ 27.14% in the w_max is achieved for the X-W_Gr beam with the graphene content of 1.5 wt% in comparison with that of the pure Cu beam. This beam exhibits the best bending resistance capability among the three graphene distribution patterns, as displayed in Fig. 12A. Furthermore, the flexibility of the beam increases when more vacancy defect is introduced into the graphene, leading to its larger bending deformation (Fig. 12B). From Fig. 12C, we can see that the maximum deflections of composite beams further increase when the beams are exposed to high temperatures. Tables 4, 5 and 6 also compare the w_max of the beams in different cases.

Fig. 12

Effects of A graphene contents, B graphene defect percentages, and C temperatures on the dimensionless maximum deflection w_max of functionally graded defective graphene reinforced composite beams

4 Conclusions

In conclusion, data-driven micromechanics models have been developed based on MD simulations and GP algorithm for predicting thermo-elastic properties of vacancy-defective graphene/Cu nanocomposites at different temperature conditions. Our simulation results suggest that Young’s modulus of composites with graphene content of 14.47 vol% at 300 K can be tailored appropriately between ~ 155.44 GPa and ~ 149.89 GPa with the vacancy defect percentage varying from 0.0% to 1.0%. The thermo-elastic properties including Young’s modulus, CTE, Poisson’s ratio, and density with their corresponding features such as graphene defect percentage, graphene content, and temperature have been collected for training based on GP algorithm to obtain the data-driven micromechanics models that can predict the properties with high R² (> 0.9) and low RMSE (< 3%). Furthermore, the thermal buckling, elastic buckling, free vibration, and static bending behaviors of functionally graded defective graphene reinforced composite beams have been investigated on the basis of the material parameters estimated by the data-driven Halpin–Tsai model and rule of mixture. Analytical results demonstrate that the functionally graded composite beam with more graphene distributed close to the surface layers exhibits a better load-carrying capability that increases with the increasing graphene content. On the other hand, the presence of vacancy defects in graphene negatively affects the structural behaviors of the composite beams to a certain degree. Results also show that the structural stiffness decreases when the beam is in high-temperature conditions, leading to decreased thermal buckling, elastic buckling, free vibration, and static bending performances.

Appendix

$$\left[ {{\text{H}}(\theta )} \right] = \left[ {\begin{array}{*{20}c} {\frac{{b_{11} \beta }}{{a_{11} }}\theta } & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ \theta & 0 & 1 & 0 & 0 & 0 \\ {\frac{{b_{11} \beta }}{{a_{11} }}\theta \cos \theta } & { - \frac{{b_{11} \beta }}{{a_{11} }}\theta \sin \theta } & 0 & 0 & 1 & 1 \\ {\sin \theta } & {\cos \theta } & 1 & 1 & 0 & 0 \\ {\theta \cos \theta } & { - \theta \sin \theta } & 1 & 0 & 0 & 0 \\ \end{array} } \right],$$

(33)

$$\left[ {{\text{H}}(\gamma )} \right] = \left[ {\begin{array}{*{20}c} {\frac{{b_{11} \beta }}{{a_{11} }}\gamma } & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ \gamma & 0 & 1 & 0 & 0 & 0 \\ {\frac{{b_{11} \beta }}{{a_{11} }}\gamma \cos \gamma } & { - \frac{{b_{11} \beta }}{{a_{11} }}\gamma \sin \gamma } & 0 & 0 & 1 & 1 \\ {\sin \gamma } & {\cos \gamma } & 1 & 1 & 0 & 0 \\ {\gamma \cos \gamma } & { - \gamma \sin \gamma } & 1 & 0 & 0 & 0 \\ \end{array} } \right],$$

(34)

$$\left[ {{\text{H}}(\lambda )} \right] = \left[ {\begin{array}{*{20}c} {\frac{{b_{11} \beta }}{{a_{11} }}\lambda } & 0 & {\frac{{b_{11} \beta }}{{a_{11} }}\lambda } & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ \lambda & 0 & \lambda & 0 & 0 & 0 \\ {\frac{{b_{11} \beta }}{{a_{11} }}\lambda \cos \lambda } & { - \frac{{b_{11} \beta }}{{a_{11} }}\lambda \sin \lambda } & {\frac{{b_{11} \beta }}{{a_{11} }}\lambda \cosh \lambda } & {\frac{{b_{11} \beta }}{{a_{11} }}\lambda \sinh \lambda } & 1 & 1 \\ {\sin \lambda } & {\cos \lambda } & {\sinh \lambda } & {\cosh \lambda } & 0 & 0 \\ {\lambda \cos \lambda } & { - \lambda \sin \lambda } & {\lambda \cosh \lambda } & {\lambda \sinh \lambda } & 0 & 0 \\ \end{array} } \right].$$

(35)