1 Introduction

Recently, functionally graded materials (FGMs) have been used in extremely high temperature environments such as the nuclear reactor, missiles, and spacecraft industries. The FGM is an advanced composite material made from a mixture of ceramics and metals.

Gradually varying in the volume fraction of the constituent materials makes the change of mechanical properties smooth and continuous from one surface to another. The improvement of FGMs is that they are able to withstand high-temperature gradient environments while their structural integrity is not destroyed. Therefore, there are many scientists interested in these novel materials. FGM cylindrical shells have attracted much attention from scientists in the world.

For FGM cylindrical shells without reinforcement stiffeners, there are many significant studies^[1-9]. For stiffened cylindrical shells, many authors focused on the behavior of these structures^[10-21]. For stiffened structures under mechanical and thermal loads, some studies have included^[22-26].

Sandwich structures can be widely used in~many applications ranging from building constructions, vehicle and airplane constructions, and refrigeration engineering because of their dominant bending rigidity, favorable vibration characteristics, low specific weight, and excellent fatigue properties. As a result, there are significant studies on the behavior of sandwich structures^[27-35].

Unlike the work^{[17, 20]}, this study takes into account the temperature and proposes a general sigmoid law and a general power law. In this paper, nonlinear buckling and postbuckling behaviors of FGM circular cylindrical sandwich shells reinforced by FGM stiffeners, under an axial compression on elastic foundations and in a thermal environment are investigated by an analytical approach. The novelty of this study is that both the shell and the stiffeners are FGMs, especially the material properties of shell are graded continuously in the thickness direction according to the general sigmoid and power laws. The thermal elements in the shell and stiffeners in equations of N_ij and M_ij are considered. The three-term solution of deflection taking into account the linear buckling shape sin (mπx/L) sin (ny/R) as well as the nonlinear buckling shape sin²(mπx/L) is more correctly selected. Using the Galerkin method, the closed-form expressions to determine critical buckling loads and postbuckling load-deflection curves are obtained. The average end-shortening ratio of shell and two cases of stiffener arrangement are considered. The influence of various parameters such as thermal parameters, the ratio between face sheet thickness and total thickness, foundation parameters, buckling modes, dimensional parameters, volume fraction indexes, and the number of stiffeners on the stability of shell is analyzed in detail.

2 Eccentrically stiffened FGM sandwich cylindrical shells

Consider a thin circular cylindrical sandwich shell (see Fig. 1) with the mean radius R, the total thickness h, and the length L only under the uniform axial compression load with the intensity p on elastic foundations in a thermal environment. Assume that two butt-ends of the shell are only deformed in their planes and they keep circular^{[3, 36]}. The coordinates x, y, and z are referred to the middle surface of the shells. Also, suppose that the shell is reinforced by closely spaced ring stiffeners and stringer stiffeners attached to inside or outside of the shell surface. The sandwich cylindrical shell is constructed from two face layers separated by a core layer made of homogeneous material or FGM. The thickness of each face layer is h_f, and the thickness of the core layer is h_co.

In this study, the FGMs of shell and stiffeners are assumed to change continuously in the thickness direction of the shell. Herein, the indexes c, m, sh, s, and r refer to ceramic, metal, shell, longitudinal stringers, and circular ring, respectively, k, k₂, and k₃ are the volume fraction indexes of shell, stringer, and ring, respectively, and k≥ 0, k₂ ≥ 0, k₃ ≥ 0. h_s and h_r denote the thickness of the stringer and the ring of the shell, respectively, and E_c, α_c and E_m, α_m are Young's moduli and the coefficients of thermal expansion of the ceramic and metal, respectively, E_cm=E_c-E_m=-E_mc, and α_cm=α_c-α_m=-α_mc. Assume that Poison's ratios υ_sh=υ_s=υ_r=υ=const.

Four FGM sandwich cylindrical shell configurations (see Fig. 2) and four rules of mixtures are considered as follows:

Type 1 FGM/metal/FGM (see Fig. 2(a))

(1)

Type 2 FGM/ceramic/FGM (see Fig. 2(b))

(2)

Types 1 and 2 are based on the general sigmoid law. When the thickness of core layer h_co= 0, from Eq. (2) , we obtain the well-known sigmoid law as in Ref. ^[37].

Type 3 Ceramic/FGM/metal (see Fig. 2(c))

(3)

Type 4 Metal/FGM/ceramic (see Fig. 2(d))

(4)

Types 3 and 4 are based on the general power law. When the thickness of face layer h_f= 0, from Eq. (4) , we obtain the well-known power law as Refs.[3]-[4] and [19]-[20].

Assume that the shell is stiffened by FGM stringers and rings. In this paper, four models with eight cases are proposed as follows:

Model 1

Case 1 Sandwich cylindrical shell of Type 1 and inside CM-stiffeners

Case 2 Sandwich cylindrical shell of Type 1 and outside MC-stiffeners

Model 2

Case 3 Sandwich cylindrical shell of Type 2 and inside MC-stiffeners

Case 4 Sandwich cylindrical shell of Type 2 and outside CM-stiffeners

Model 3

Case 5 Sandwich cylindrical shell of Type 3 and inside MC-stiffeners

Case 6 Sandwich cylindrical shell of Type 3 and outside MC-stiffeners

Model 4

Case 7 Sandwich cylindrical shell of Type 4 and inside CM-stiffeners

Case 8 Sandwich cylindrical shell of Type 4 and outside CM-stiffeners

Here, CM is used to denote the material properties varying from ceramic to metal, and MC denotes the material properties varying from metal to ceramic.

In order to guarantee the continuity between the FGM sandwich shell and FGM stiffeners, the material properties of stiffeners are defined as follows^[19-20]:

For inside CM-stiffeners,

(5)

and for outside CM-stiffeners,

(6)

Similarly, for inside and outside MC-stiffeners, Young's moduli and thermal expansion coefficients are given in Appendix A.

3 Governing equations

Based on the nonlinear strain-displacement relations of sandwich cylindrical shells, the mid-surface strain components of the shell are expressed in the form of^{[36, 38-39]}

(7)

where u=u(x, y), v=v(x, y), and w=w(x, y) are the displacement components of the middle surface points with respect to x-, y-, and z-axes, respectively.

The strain components across the shell thickness at a distance z from the mid-surface are given by

(8)

in which k_x, k_y, and k_xy are the change of curvatures and twist of the shell, respectively.

Using Eq. (7) , we obtain the compatibility equation as follows:

(9)

The stress-strain relationship based on the Hooke's law is defined as follows:

For the shell,

(10)

and for stiffeners,

(11)

where T₀ is the room temperature.

The contribution of stiffeners is taken into account using the smeared stiffener technique. The expressions for force and moment resultants of an eccentrically stiffened FGM sandwich cylindrical shell are given by^{[19-20, 38]}

(12)

(13)

where the stiffness parameters C_ij are given in Appendix A.

The reverse relations are obtained from Eq. (12) as

(14)

Substituting Eq. (14) into Eq. (13) yields

(15)

where C_ij^* and D_ij^* are defined in Appendix B.

According to the classical shell theory, the equilibrium equations of sandwich cylindrical shell taking into account elastic foundations are given by^{[3, 38-39]}

(16)

where K₁ (N/m³) is the Winkler foundation modulus, and K₂ (N/m) is the shear layer stiffness of the Pasternak model.

Introduce a stress function φ (x, y),

(17)

It can be seen that the first two equations of Eq. (16) are identically satisfied. Substituting Eqs. (15) and (17) into the third equation of Eq. (16) and using Eq. (8) , we obtain

(18)

where

(19)

Equation (18) includes two dependent unknown functions w and φ, and to find a second equation relating to these two functions, the geometrical compatibility equation (9) is used. For this purpose, introducing Eq. (14) into Eq. (9) leads to

(20)

where

(21)

Equations (18) and (20) are two nonlinear governing equations used to analyze the stability of stiffened FGM sandwich cylindrical shells under an axial compression load on elastic foundations in a thermal environment.

4 Buckling and postbuckling analysis

Consider the sandwich cylindrical shell with simply supported boundary conditions at the edges x=0 and x=L. Then, the deflection of axially loaded shell satisfying boundary condition on the average sense and stress function takes the form of^{[3, 36]},

(22)

where α=mπ/L, β=n/R, m and n are the half wave number along the x-axis and the wave number along the y-axis, respectively, σ_0y is the negative average circumferential stress, and

(23)

in which the coefficients a_i are defined as

(24)

Introduce w and φ into the left side of Eq. (18) . Then, applying the Galerkin method in the ranges 0≤y≤2πR and 0≤x≤L, we obtain

(25)

(26)

(27)

where

(28)

In addition to Eqs. (25) -(27) , the sandwich cylindrical shell must also satisfy the circumferential closure condition^{[3, 40]}, that is,

(29)

Using Eqs.(14), (17), and (22) , this integral gives

(30)

4.1 Shells subjected to combined axial mechanical and thermal loads

This section considers shells subjected to combined axial mechanical and thermal loads. First, from Eq. (25) , we have

(31)

Substituting Eqs. (26) and (31) into Eq. (30) yields

(32)

With f₀ just found, Eq. (26) becomes

(33)

Equation (31) is rewritten as

(34)

where

(35)

Finally, substituting Eqs. (32) -(34) into Eq. (27) , we obtain

(36)

The expression (36) is used to determine the critical loads and to analyze the postbuckling load-deflection curves of nonlinear buckling shape of stiffened FGM sandwich cylindrical shells under an axial load on elastic foundations and in a thermal environment.

From Eq. (36) , taking f₂→0, we obtain the expression of upper buckling compressive load as

(37)

Assume that the temperature change is uniformly raised from the initial value T_i to the final one T_f, i.e., △T=T_f -T_i=a, where a is a constant. Then, the thermal parameters and are calculated in (A5) , (A7) , and (A9) .

4.2 Maximal deflection

From Eq. (22) , it is observed that the maximal deflection of the shells

(38)

locates at x=iL/(2m), y=jπR/(2n), where i and j are odd integer numbers.

Substituting Eqs. (32) and (33) into Eq. (38) , we have

(39)

Combining Eq. (36) with Eq. (39) , the effects of material and geometrical parameters on the postbuckling load-maximal deflection curves of sandwich shells will be analyzed in numerical calculations.

4.3 Average end-shortening ratio △_x

The average end-shortening ratio △_x is defined by^{[3, 36]}

(40)

Using Eqs. (14) , (17) , and (22) , this integral gives us

(41)

Substituting Eqs. (33) and (34) into Eq. (41) leads to

(42)

Combining Eq. (36) with Eq. (42) , the effects of inhomogeneous and dimensional parameters on postbuckling load-average end-shortening ratio curves of sandwich shells with f₁ ≠ 0 and f₂ ≠ 0 can be analyzed.

In the case of f₁ =0 and f₂ =0, Eq. (41) becomes

(43)

Using Eqs. (33) and (34) , Eq. (43) becomes

(44)

Equation (44) shows that when f₁ =0 and f₂ =0, the relation between △_x and p is linear. Combining Eq. (36) with Eqs. (42) and (44) , the effects of inhomogeneous and dimensional parameters on postbuckling load-average end-shortening ratio curves of sandwich shells will be investigated.

4.4 Sandwich shell only subjected to mechanical load (△T=0)

If the shell is only subjected to a mechanical load (△T=0), from Eq. (36) , we obtain

(45)

Equation (45) is used to determine the critical mechanical load and to analyze the postbuckling load-deflection curves of nonlinear buckling shape of stiffened FGM sandwich cylindrical shells only under an axial load on elastic foundations.

From Eq. (45) , taking f₂ → 0 leads to the upper mechanical buckling load p_upper of shell without the thermal element as follows:

(46)

5 Numerical result and discussion 5.1 Validation of present approach

To verify the present study, three comparisons on the critical load are carried out with the results from the literature.

Comparison 1 Table 1 (with △T=0 and Eq. (46) ) compares the results of present critical buckling load of un-stiffened isotropic cylindrical shells under an axial compression with the results of Brush and Almroth^[38] (based on equation (5.50) in page 168) .

Comparison 2 Table 2 (with △T=0 and Eq. (46) ) compares the critical buckling axial load of un-stiffened FGM cylindrical shells with the results of Huang and Han^[40].

Comparison 3 Table 3 compares the results on the critical buckling load of stiffened isotropic homogeneous cylindrical shells with the result of Brush and Almroth^[38] (based on equation (5.78) in page 180) and with the result of Bich et al.^[17].

It can be seen that the results of three comparisons affirm the validity of the present approach.

In each subsection below, to illustrate the present approach for nonlinear buckling and postbuckling analysis of stiffened FGM sandwich cylindrical shells of Type 4 under an axial compression, consider a ceramic-metal shell consisting of Alumina and Aluminum with the following properties E_c=380 GPa, α_c =5.4× 10^-6 1/K, E_m =70 GPa, α_m=22.2× 10^-6 1/K, and υ = 0.3. The material and geometrical parameters of shell are taken as k=1, h=0.006 m, R= 0.6 m, L=1.2 m, K₁=2.5×10⁷ N/m³, K₂=2.5×10⁵ N/m, h_s=h_r=0.006 m,b_s=b_r=0.006 m, and n_s=n_r=20.

5.2 Effect of ratio between face sheet thickness and total thickness h_f/h

Table 4 shows the effect of the ratio between face sheet thickness and total thickness h_f/h on the upper critical load p_cr (GPa) of stiffened FGM sandwich cylindrical shells (Type 4) on elastic foundations with k=0.2, 0.5, 1, 5, 10 for inside and outside stiffeners. It can be seen that the critical load decreases as the ratio h_f/h increases. For example, with k=1, the inside stiffeners p_cr = 1.349 7 GPa (h_f/h=0.1) decreases about 3.41% in comparison with p_cr= 1.305 2 GPa (h_f/h=0.2) .

Table 5 compares the critical upper load of different types of stiffened FGM sandwich cylindrical shells on elastic foundations with the change of the ratio h_f/h. It can be seen that the critical loads decrease with the increased ratio h_f/h for Types 2, 3 and 4. Conversely, the critical thermal load increases with the increased ratio h_f/h for Type 1.

5.3 Effects of mechanical load

Table 6, using Eq. (45) and △T=0, shows the effects of buckling mode (m, n) on the critical lower buckling load of shell. In this case, the critical lower buckling load is p_lower=595.356 2 MPa corresponding to (m, n) = (3, 6) .

Figure 3, using Eqs. (45) and (39) , presents the effects of volume fraction index k = 0, 0.5, 1 on the p-W_max/h nonlinear postbuckling curves. The curves become lower gradually when k increases. This is reasonable with the characteristic material, because the increase of k corresponds to purer ceramic. Figure 3 also shows the value of the lower critical buckling load mentioned in Table 6.

Figure 4 examines the variation of p versus △_x with various combinations of mode (m, n) by using Eqs. (42) , (44) , and (45) . It is observed that at the first stage, p increases linearly versus △_x until the linear bifurcation point is reached. At the next stage, it follows the postbuckling equilibrium path in which a postbuckling mode is continuously jumping. The carrying capacity of the shell in this stage decreases strongly until the lowest point and then increases gently. The interesting similar characteristic on postbuckling mode jumps of un-stiffened cylindrical shell was also noted by Huang and Han^[4].

Table 7 shows effects of k on the upper critical load. It is found that the upper critical load decreases with the increase of k and it is the greatest with k=0. In addition, Table 7 also shows that the bigger the foundation parameters K₁ and K₂ are, the greater the value of upper critical load is. Especially, the upper critical load corresponding to the presence of both foundation parameters K₁ and K₂ is the biggest. The upper critical load of shell without foundation is the smallest.

Once again, the effects of k on load-carrying capacity of shells are illustrated in Figs. 5, 6, and 7. It is found that the critical buckling load decreases with the increase of k. This property is suitable for the real property of material, because the value of k= 0 corresponds to a ceramic-richer shell which has higher stiffness than a metal-richer one.

Table 8 shows the effects of R/h on the upper critical load. The upper critical load pupper decreases markedly with the increase of R/h.

This result agrees with the actual property of structure because the thinner the shell is, the smaller the value of critical load is. This remark is also illustrated in Fig. 7 and Table 9. In addition, the obtained result indicates that the buckling load p_cr decreases with the increase of L/R.

5.4 Effects of thermal and mechanical loads

Using Eqs. (36) and (37) , with the data given in Subsection 5.3, the effects of stiffeners, foundation, volume fraction index, geometrical parameter, and temperature increase △T on critical buckling and postbuckling loads are analyzed. Figure 8 indicates the effects of the temperature increase △T on postbuckling p-W_max/h curves. Based on Fig. 8 and Tables 10-12, the critical load of the shell decreases when △T increases.

Table 9 shows that the buckling load increases continuously with the increased number of stiffeners. For example, for inside stiffened shells, the critical load of p_lower = 635.481 9 MPa (n_s=n_r=30) increases about 9.6% in comparison with the critical load of p_lower= 574.572 7 MPa (n_s=n_r=15) . Also, it can be seen that the critical load pcr of inside stiffened shells is smaller than that of outside stiffened shells.

Figure 9 demonstrates the effects of temperature increment △T on postbuckling curves _p-△_x. Once again, it shows that the critical load of the shell decreases when △T increases. Furthermore, when the temperature increment △T is not equal to 0 K, the starting point has negative average end-shortening ratio △_x, because the shell hatches. If △T = 0 K, the postbuckling curve starts at original. This remark is also indicated in Fig. 10. Figure 10 presents the effects of temperature increment △T and R/h ratio on postbuckling curves _p-△_x . This figure shows that when R/h increases, the postbuckling curves are significantly lowered.

Tables 10 and 11 indicate that the critical loads of shells with outside stiffeners are the biggest, those of shells with inside stiffeners are smaller, and those of un-stiffened shells are the smallest. In addition, the critical loads decrease when one of the parameters △T, R/h, L/R increases.

Table 12 shows that the critical load of eccentrically stiffened functionally graded circular cylindrical shells on elastic foundations in a thermal environment increases with the increased number of stiffeners and foundation parameters K₁ and K₂.

In Fig. 11, the postbuckling p-W_max/h curves of shells are traced by Eqs. (39) and (42) . It can be seen that the the postbuckling curves are lower when R/h increases.

Figure 12 presents the effects of temperature increase △T on p-k curves. The lower critical load p_cr of eccentrically stiffened FGM shells on elastic foundations in a thermal environment reduces when k increases.

6 Conclusions

An analytical investigation is presented to analyze nonlinear buckling and postbuckling behaviors of eccentrically stiffened FGM sandwich cylindrical shells under an axial compression, resting on elastic foundations and in a thermal environment. Two cases of rings and stringers attached eccentrically to the inside and outside of shell are considered. Four cases of the general sigmoid law and general power law are proposed. The thermal elements of shell and stiffeners in equations of N_ij and M_ij are taken into account. Some improvements and highlights on theoretical and scientific values are obtained as follows.

(ⅰ) The expression of deflection with three-term including linear and nonlinear buckling shapes is more correctly chosen.

(ⅱ) The closed-form expressions to determine critical buckling loads and nonlinear postbuckling load-deflection curves are obtained.

(ⅲ) Input parameters including temperature, stiffeners, core layer, foundation, geometrical dimensions and volume fraction index significantly affect buckling and postbuckling maximal deflection-load curves of the shell.

(ⅳ) An average end-shortening ratio of the shell is considered.

Appendix A

In Eqs. (12) and (13),

(A1)

For the sandwich cylindrical shell of Type 1,

(A2)

For the sandwich shell of Type 2, the expressions of E_i (i = 1, 2, 3) are similar to those of E_i (i = 1, 2, 3) of the sandwich plate of Type 1 by replacing E_cwith E_m and replacing Emc with E_cm. For the sandwich plate of Type 3,

(A3)

For the sandwich shell of Type 4, the expressions of E_i (i = 1, 2, 3) are similar to those of E_i (i = 1, 2, 3) of the sandwich plate of Type 3 by replacing E_c with E_m and replacing E_mc with Ecm.

(A4)

If △T =const., then with

(A5)

For inside CM-stiffeners,

(A6)

If △T =const., then with , and

(A7)

For outside CM-stiffeners,

(A8)

If △T =const., then with and

(A9)

in which b_s and b_r are the widths of stiffeners. Also, d_s and d_r denote the distances between two stringers and rings, respectively (see Fig. 1).

For inside and outside MC-stiffeners, the expressions are similar to those for inside and outside CM-stiffeners by replacing c by m and replacing m by c.

Appendix B

In Eqs. (14) and (15),

(B1)

(B2)