Higher-order vibration of thick composite and sandwich plates based on an alternative higher-order model

Jinghui DENG, Tngzhen WU, Zhen WU, Zhengling LIU, Xiohui REN

a China Helicopter Research and Development Institute, Jingdezhen 333001, China

b School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China

c School of Mechanical Engineering, Xi’an Aeronautical University, Xi’an 710065, China

KEYWORDS Analytical solution;Higher-order theory;Natural frequency;Sandwich plate;Transverse stretching vibration

Abstract The transverse stretching vibration of thick sandwich plates, which is attributed to largely different stiffness at the adjacent layers, is a challenging issue, and efficient approach for such issue is less reported in the published literature.Thus,natural frequencies corresponding to stretching vibration modes are generally neglected in engineering design, which might impact structural safety as frequencies of the exciting force are close to transverse stretching vibration frequencies.This paper proposes an alternative higher-order model for dynamic analysis corresponding to the higher-order vibration modes.The proposed model is classified in the displacement-based equivalent single-layer theory, as the number of displacement parameters in the proposed model is independent of the layer number.The continuity of displacements and transverse shear stresses can be fulfilled at the interfaces between the adjacent layers of structures.To demonstrate the capability of the proposed model, typical examples are analyzed by utilizing the proposed model, the threedimensional finite element method and the chosen higher-order models.By comparing with the exact three-dimensional elasticity solutions, it is found that the proposed model can yield more accurate natural frequencies corresponding to the higher-order displacement modes than the selected models.Moreover, the factors influencing reasonable prediction of the higher-order frequencies are investigated in detail, which can provide a reference for the accurate prediction of the higher-order frequencies.

1.Introduction

Owing to their high stiffness and strength ratio,the composite and sandwich structures are widely utilized in aerospace, ship and mechanical engineering.1Compared to conventional structures, numerous complicated mechanical behaviors arise in the design,analysis and manufacturing of layered composite and sandwich structures.Therefore, the development of twodimensional theories is fundamental for the analysis of layered composite structures.2As a result, a number of twodimensional plate theories approximating the threedimensional solution have been developed.

Based on the first-order shear deformation theory, the thermal vibration behaviors of thick composite plates have been investigated.3Ngo-Cong et al.4utilized the first-order shear deformation theory to analyze free vibration of layered composite plates.In order to avoid using the shear correction factors, Reddy5proposed a third-order theory including five independent variables.Based on the Reddy’s model, two finite element formulations have been constructed for free vibration analysis of laminated composite and sandwich plates.6By utilizing the sinusoidal shear function, Le′vy7and Touratier8proposed the sinusoidal shear deformation theory.In the light of the sinusoidal model, Zenkour9studied the hygrothermal influences on bending behavior of angle-ply laminated plates.Zenkour et al.10also investigated thermal bending of cross-ply composite plates resting on the elastic foundations by using the sinusoidal model.By using an intelligent design, Zenkour11developed a higher-order model containing transverse normal strain, and only four independent variables were involved in the proposed model.Furthermore,the four-unknown model was extended to hygrothermal behavior of antisymmetric composite plates.12By means of the four-unknown higher-order model, Thai et al.13studied dynamic behavior of the functionally graded isotropic and sandwich plates.Recently, the four-unknown model was employed to predict stresses of functionally graded plates containing piezoelectric faces.14,15Wu et al.16developed an advanced five-unknown higher-order theory to study dynamic behavior of composite and sandwich plates.

Carrera17attempted to unify the higher-order shear deformation theories in a unique formulation,and the Unified Formulation (UF) was widely used to assess transverse shear stresses of the composite and sandwich plates.18In the light of the unified formulation, Xu et al.19investigated the free vibration of open thin-walled beams.Foroutan et al.20employed the unified formulation to study the post-buckling and large-deflection behaviors of the sandwich plates with functionally graded core.Numerous cases of post-buckling problems were studied to illustrate the capability of the unified formulation.Carrera and Demirbas21extended the unified formulation to study the bending and post-buckling response of thin-walled functionally graded beams.Le et al.22constructed a beam element based on the third-order theory to investigate free vibration and buckling behaviors of functionally graded sandwich beams.In the light of the higher-order shear deformation model,Vinh23predicted the stresses and displacements of bi-directional functionally graded sandwich plates.Wu et al.24developed a sinusoidal global–local theory to investigate dynamic behavior of sandwich plates, and the proposed model was verified by testing experiments.

By introducing the local zig-zag function in the in-plane displacement field,Di Sciuva25proposed a zig-zag model to study mechanical behavior of laminated composite structures.Di Sciuva and Gherlone26also proposed a third-order Hermitian zig-zag model to investigate mechanical behavior of layered structures subjected to the tangential loads on the surfaces.Cho and Parmerter27proposed a higher-order zig-zag model for mechanical analysis of laminated composite structures.By considering transverse normal strain, Cho and Oh28proposed a refined higher-order zig-zag theory to analyze the deformation and stresses of thick layered plates under mechanical, thermal and electric loads.Based on the global and local displacement assumption,29Wu and Chen30proposed a global–local higher-order theory containing transverse normal strain to investigate mechanical behavior of laminated composite structures.In order to accurately predict the crack propagation of laminated composite structures, Zhang et al.31developed a new phase field method.Moreover, the good performance of the proposed model was verified by the experimental and other numerical results.Hirshikesh et al.32utilized the phase field method to study the crack propagation in the laminated composite structures with variable stiffness.By combining the phase field model and the cohesive element,Zhang et al.33constructed a computational framework to investigate the progressive failure in composite structures.Moreover, the proposed method was implemented into commercial software ABAQUS.In addition, Bui and Hu34presented a critical review on the developments and the applications of the phase field models in laminated composite structures.

By reviewing the literature, it can be found that a number of higher-order models have been developed to study static and dynamic behavior of the functionally graded,composite and sandwich structures.Nevertheless, in-plane displacement field of the proposed model generally includes the third-order polynomial, and transverse normal strain is also discarded.Previous investigations35show that the third-order models containing transverse normal strain might also lose the capability to accurately predict natural frequencies corresponding to the higher-order displacement modes.However, fewer investigators pay attention to such issues.As a result, these frequencies corresponding to the higher-order displacement modes are usually neglected in structural design of aircraft, which might have a significant influence on structural safety in aircraft.Therefore, this paper proposes an alternative higher-order model to predict accurately natural frequencies corresponding to the higherorder displacement modes.The three-dimensional elasticity solutions, three-dimensional finite element results and the results obtained from other higher-order models have been selected to evaluate the capability of the proposed model.Moreover, the influences of various parameters on natural frequencies corresponding to higher-order displacement modes are also investigated in detail.

This paper proposes an alternative higher-order model containing the thickness-stretching effects,so that natural frequencies corresponding to the higher-order displacement modes can be reasonably predicted by using the proposed model.Firstly,free vibration of a four-ply plate is investigated, and performance of the proposed model is also verified by comparing to exact solutions.Subsequently, the proposed model is utilized to study free vibration of square and rectangular sandwich plates, where three types of vibration modes will be analyzed in detail.In addition, influence of transverse normal strain on the frequencies corresponding to the higher-order modes will be illustrated detailedly.Moreover, it is expected that accuracy of natural frequencies corresponding the higher-order modes can be improved by using the proposed model.

2.An alternative Higher-Order Theory Including Thickness-Stretching Effect (HTIT)

In view of computational efficiency and accuracy, the models containing the third-order polynomial are widely selected for mechanical analysis of the composite and sandwich structures.For analysis of thick composite structures, the second-order polynomial is also employed to simulate the thicknessstretching effect.28,36Nevertheless, these models might fail to produce accurate natural frequencies with respect to the higher-order transverse displacement modes of the thick composite and sandwich plates.Furthermore, such issue is generally discarded, which will influence structural safety.As a result, an alternative higher-order theory including thicknessstretching effect is developed to accurately yield natural frequencies corresponding to the higher-order transverse displacement modes of the thick composite and sandwich plates.The initial displacement field for thick composite and sandwich plates can be expressed as

When the local displacements are introduced in the inplane displacement field, the continuity conditions at the adjacent layers of the in-plane displacement component will be destroyed, which will significantly influence the accuracy of the model.Thus, the continuity condition of in-plane displacements at the interfaces will be enforced in advance.By utilizing the continuity condition of in-plane displacements between the kth layer and the (k–1)th layer, Eq.(3) can be written:

where Aiand Bican be found in Appendix A, i = 1,2,...,11.

After applying the continuity conditions of in-plane displacement and transverse shear stress at the interfaces as well as the free surface condition,all local displacement parameters in the initial displacement field have been constrained, so final displacement field of the proposed model can be given by

where φiand φican be seen in Appendix A.

Eq.(14) shows that all local displacement parameters have been removed from the starting displacement field.

3.Analytical solutions for simply-supported composite plates

3.1.Constitutive equations

The transverse normal strain has been taken into account in the proposed model,so that the three-dimensional constitutive equation with reference to the material coordinate axes can be written as

where Qijis the transformed elastic constants corresponding to the global coordinate axes.

In the light of the proposed model, the strain components for the composite plate can be presented by

3.2.Hamilton’s principle

3.3.Analytical solution

In the light of Eq.(19),the analytical solution for the cross-ply composite and sandwich is presented by utilizing Navier’s solution procedure.37According to Navier’s solution technique, displacement parameters satisfying the simplysupported boundary conditions can be written as

where uimn, vimnand wimnare the generalized displacement parameters, which can be found in Eq.(20).

The circular frequencies of composite plates can be acquired by solving the following eigenvalue problem of the simply-supported composite plates:

4.Numerical results and discussion

Two typical examples are selected to investigate the effect of transverse stretching deformation on the dynamic behavior of layered composite and sandwich plates.In order to illustrate the accuracy of the proposed model,three-dimensional elasticity solutions2,38have been recalled to evaluate the capability of the proposed model.Nevertheless, transverse stretching displacement modes based on the three-dimensional elasticity analysis are less reported in the literature.Therefore, Three-Dimensional Finite Element Model (3D-FEM) of ABAQUS is employed to produce transverse stretching displacement modes of composite and sandwich plates.In addition, several higher-order models recently published in literature are also chosen for comparison.

4.1.Example 1: Free vibration of a four-layer plate [0°/90°/0°/90°] is carried out to assess capability of proposed model

Free vibration of a typical four-layer composite plate is selected to evaluate the capability of the developed model,and the effect of transverse normal strain on vibration modes of laminated composite plates is also investigated in detail.As the half-wave number m = 1 and n = 1, it is found that natural frequencies acquired from the proposed model HTIT are in exact agreement with the exact solutions (Exact)38in Table 1,28,35,38because the maximum percentage error of the present results relative to the exact solutions is merely 0.68 %.Fig.128,35,38clearly shows that the line of the present results has covered that of the exact solutions.The higherorder Zig-Zag Theory Containing Transverse Normal Strain(ZZTT)28is chosen to assess the proposed model.Numerical results show that the model ZZTT can only produce seven frequencies,as seven independent variables are only contained in the model ZZTT.Furthermore,the maximum percentage error of results of the ZZTT corresponding to exact solutions reaches 30.01 %.

Fig.1 Comparison of higher-order frequencies for a four-layer composite plate (a/h = 10, m = 1, n = 1).28,35,38

Fig.2 Comparison of higher-order frequencies for a four-layer composite plate (a/h = 4, m = 1, n = 1).28,35

Table 1 Comparison of present results to exact solution and other published results for a four-ply plate[0°/90°/0°/90°] (a/h = 10,m = 1, n = 1).28,35,38

Table 2 Comparison of present results to exact solution and other published results for a four-ply plate [0°/90°/0°/90°] (a/h = 10,m = 2, n = 1).28,35,38

Fig.3 Comparison of higher-order frequencies for a four-layer composite plate (a/h = 10, m = 2, n = 1).28,35,38

In addition, five models in Ref.35 have been chosen for comparison, and the acronyms of these models will be illustrated in detail.Herein, LW4 represents the layerwise model,where the fourth-order function at each ply is used.M or D signifies the mixed or the classical analysis by means of displacement formulation,and the number 3 represents the order number of stress and displacement fields with respect to thickness coordinate z.In addition, the characters i and d indicate that transverse normal strain is contained and neglected,respectively.It can be seen that the model M3i proposed by Carrera35can produce more accurate results than other chosen models, as transverse normal strain has been considered andthe continuity conditions of interlaminar stresses have been enforced.These results show that transverse normal strain should be taken into account to accurately yield the higher frequencies of laminated composite plates.In Fig.2,28,35it is noted that the first six frequencies of a thick composite plate(a/h = 4) can be reasonably produced by using the models HTIT, ZZTT and D3i.However, other frequencies have been overestimated by the models ZZTT, D3i and D3d.

Table 4 Boundary conditions of 3D finite element model (3D-FEM) with four simply-supported boundaries.

As the half-wave number m = 2 and n = 1, the results acquired from the chosen models are shown in Table 228,35,38Again, it is noticed that the maximum percentage error of the present results relative to the exact solutions is 0.77 %,which shows that the proposed model can also produce the higher-order frequencies accurately with the increase of halfwave numbers.Moreover, all results acquired from the selected models are plotted in Fig.3,28,35,38where the results obtained from the proposed model HTIT and the model LW4 nearly cover the exact solutions.

Fig.5 Percentage errors of all results relative to exact solutions for five-ply sandwich plate made of composite face sheets and PVC core [0°/90°/C/90°/0°] (Mode I, a/h = 5, tc = 0.7h,ts = 0.15h).5,28,40

Fig.4 Comparison of displacement modes acquired from model HTIT and 3D-FEM for a five-layer sandwich plate(Mode I,a/b=1,a/h = 5, tc = 0.7h, ts = 0.15h).

Table 5 Contrast of natural frequencies for five-ply sandwich plate made of composite face sheets and PVC core [0°/90°/C/90°/0°](Mode II, tc = 0.7h, ts = 0.15h).2,28,39

Fig.6 Comparison of displacement modes acquired from model HTIT and 3D-FEM for a five-layer sandwich plate(Mode II,a/b=1,a/h = 5, tc = 0.7h, ts = 0.15h).

Fig.7 Percentage errors of all results relative to exact solutions for five-ply sandwich plate made of composite face sheets and PVC core [0°/90°/C/90°/0°] (Mode II, a/h = 5, tc = 0.7h,ts = 0.15h).28

4.2.Example 2: Free vibration of a five-ply sandwich plate [0°/90°/C/90°/0°] composed of composite face sheets and Polyvinyl Chloride (PVC) foam

Material properties of the composite face sheet and isotropic core (PVC)2are given by

(1)Face sheets:E1=132.38 GPa,E2=E3=10.756 GPa,G12=G13=5.6537 GPa,G23=3.603 GPa,v12=v13=0.24,v23= 0.49, ρs= 1600 kg/m3.

In general, the bending modes of the composite and sandwich plates are studied,which are called as the I-type displacement modes (Mode I) herein.By comparing to the material properties of the composite face sheet and isotropic core(PVC),it is observed that the material constants of face sheets are largely more than those of the PVC core, which will resultin the transverse stretching vibration modes for thick sandwich structures.Transverse stretching vibration modes are called as the II-type displacement modes (Mode II).2In addition, the third displacement mode is called as the III-type displacement modes (Mode III),2as the half-wave numbers m and n are fixed.

Table 6 Contrast of natural frequencies for five-ply sandwich plate made of composite face sheets and PVC core [0°/90°/C/90°/0°](Mode III, tc = 0.7h, ts = 0.15h).2,28,39

Table 7 Contrast of natural frequencies for five-ply rectangular sandwich plate made of composite face sheets and PVC core[0°/90°/C/90°/0°] (Mode I, b/a = 3, tc = 0.7h, ts = 0.15h).2,39

The numerical results of normalized natural frequencies acquired from the present model and the chosen models along with exact three-dimensional elasticity solutions(Exact 3D)by Brischetto2are shown in Table 3,2,5,28,39,40where tcand tsare the thickness of core and face sheet, respectively.From Table 3,2,5,28,39,40it can be observed that the present resultsand the results acquired from the model ZZTT corresponding to Mode I are in good agreement with the exact solutions.Nevertheless,the higher-order models RHSDT5and BHSDT40overestimate natural frequencies as transverse normal strain has been neglected.It is interesting that the model considering transverse normal strain(SHSDT)39loses the capability to predict natural frequencies of thick (a/h = 5) and moderately thick(a/h=10)plates.In Table 3,2,5,28,39,40number in bracket denotes element number used in 3D-FEM analysis.Boundary conditions used in the three-dimensional finite element analysis are shown in Table 4.It is shown that 4.08 s are spent to produce natural frequencies of Mode I, Mode II and Mode III(m = 1, n = 1) by using the present model HTIT.Moreover,it takes about 16.32 s to produce the frequencies corresponding to the first four half waves (m = 1, n = 1; m = 1, n = 2;m = 2, n = 1; m = 2, n = 2) by using the present model HTIT.Using the same computer, it has to take about 57 min to obtain the first four frequencies by using 314600 elements of the 3D-FEM in ABAQUS.

Table 8 Contrast of natural frequencies for five-ply rectangular sandwich plate made of composite face sheets and PVC core[0°/90°/C/90°/0°] (Mode II, b/a = 3, tc = 0.7h, ts = 0.15h).2,39

Fig.8 Comparison of displacement modes acquired from model HTIT and 3D-FEM for a five-layer sandwich plate(Mode I,b/a=3,a/h = 5, tc = 0.7h, ts = 0.15h).

In Fig.4, Mode I acquired from the present model is compared with the 3D-FEM results of ABAQUS.In addition,the percentage errors of the results relative to exact solutions for a five-ply sandwich plate made of composite face sheets and PVC core [0°/90°/C/90°/0°] (Mode I, a/h = 5, the thickness of core tc= 0.7h, the thickness of face sheet ts= 0.15h) have been plotted in Fig.5,5,28,40where it can be found that the maximum percentage error of the present results relative to exact solutions is less than 1.5 %.

Table 52,28,39presents the normalized frequencies corresponding to the II-type displacement mode.From Table 5,2,28,39it is observed that the present results are in good agreement with the exact solutions.2However, the model ZZTT including transverse normal strain loses the capability to produce natural frequencies accurately corresponding to the II-type displacement modes.In Fig.6,the II-type displacement modes corresponding to different half-wave numbers are compared to those obtained from 3D-FEM.From Figs.4 and 6,it is observed that the displacement modes corresponding to Mode II are largely different from those of Mode I.Fig.728shows that the maximum percentage error of the present results (Mode II) relative to exact solutions is less than 11 %.In Table 6,2,28,39natural frequencies corresponding to Mode III acquired from the chosen models are compared with the exact solutions.Numerical results corresponding to different half-wave numbers(m,n)show that the present results are in good agreement with the exact solution.2The models SHSDT and ZZTT can reasonably predict fundamental frequency (m = 1, n = 1) of Mode III.With the increase of half-wave numbers (m, n), the accuracy of natural frequencies acquired from the model ZZTT gradually decreases.

Fig.9 Comparison of displacement modes acquired from model HTIT and 3D-FEM for a five-layer sandwich plate(Mode II,b/a=3,a/h = 5, tc = 0.7h, ts = 0.15h).

Table 9 Contrast of natural frequencies for five-ply rectangular sandwich plate made of composite face sheets and PVC core[0°/90°/C/90°/0°] (Mode III, b/a = 3, tc = 0.7h, ts = 0.15h).2,39

In order to further assess the performance of the proposed model, the dynamic behavior of a rectangular sandwich plate(b/a=3)corresponding to the I-type,II-type and III-type displacement modes is investigated in detail.In Table 7,2,39the normalized frequencies corresponding to the I-type displacement mode have been compared with the exact solutions,3D-FEM results and those obtained from the model SHSDT.Numerical results show that the present results are in exact agreement with the exact solutions2as the maximum percentage error of the present results relative to the exact solutions is merely 1.48 %.The model SHSDT overestimates the natural frequencies corresponding to Mode I.In Table 8,2,39the present results corresponding to Mode II have been compared with the exact solutions.Again, the present results agree well with the exact solutions.Moreover, the model SHSDT also accurately predicts fundamental frequencies corresponding to Mode II.In addition, the displacement modes corresponding to Mode I and Mode II have been plotted in Figs.8 and 9.In Table 9,2,39natural frequencies corresponding to Mode III are compared to the exact solutions.Numerical results show that the present results are in good agreement with the exact solutions.Nevertheless, the model SHSDT39fails to accurately produce fundamental frequencies corresponding to Mode III for thick plate (a/h = 5).

5.Conclusions

In order to investigate the transverse stretching vibration of thick composite and sandwich plates, a higher-order theory containing transverse normal strain has been developed.Furthermore,the proposed model can fulfill the interlaminar continuity conditions and the free surface conditions of transverse shear stresses,which can improve the accuracy of the predicted frequencies.The proposed model has been verified through several examples.Numerical results show that the proposed model can produce more accurately natural frequencies corresponding to different displacement modes compared to those acquired from the chosen models.Transverse normal strain has a significant impact on prediction of frequencies corresponding to Mode II and Mode III.However, the model containing transverse normal strain and satisfying interlaminar continuity of transverse shear stresses might also fail to accurately predict natural frequencies of Mode II and Mode III.Therefore,the present model is recommended to study the free vibration behavior of thick composite and sandwich plates.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China (No.12172295), SKLLIM1902, and the Natural Science Foundation in Shaanxi Province, China (No.2019JQ-909).

Coefficients in Eq.(14) can be written as