2016, Vol. 37
Shanghai University
Article Information
- R. ANSARI, T. POURASHRAF, R. GHOLAMI, H. ROUHI. 2016.
- Analytical solution approach for nonlinear buckling and postbuckling analysis of cylindrical nanoshells based on surface elasticity theory
- Appl. Math. Mech. -Engl. Ed., 37(7): 903-918
- http://dx.doi.org/10.1007/s10483-016-2100-9
Article History
- Received 2016-01-27;
- in final form 2016-03-28
2. Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, Lahijan 1616, Iran;
3. Department of Engineering Science, Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rudsar-Vajargah 44891-63157, Iran
Theoretical investigations on the behavior of nanostructures are generally performed via atomistic simulations and continuum models. Although the accuracy of atomistic simulation is very high, their computational cost is noticeably higher than that of continuum approaches especially for nanostructures with a large number of atoms. As a result, continuum models as computationally efficient tools are extensively used for modeling nanostructures with a large number of atoms. Classical continuum models are size-independent, making their applicability to nanostructures questionable. As the size of nanostructures is scaled down to very small scales, size effects play an important role in their mechanical behavior. Thus, classical continuum models need to be extended in order to take the size effects into account. Up to now, several nonclassical continuum theories have been developed for considering the small scale effects in the analysis of micro- and nanostructures. The nonlocal[1, 2], couple stress[3, 4, 5], strain gradient[6, 7, 8], micropolar, and micromorphic[9, 10] theories can be mentioned for example.
The nonlocal effect is one of the size effects which cannot be taken into account by the classical continuum mechanics. Such an effect is due to the discrete nature of matter and can be captured based on the Eringen nonlocal elasticity theory[1, 2]. There are several research works in the literature on the mechanical problems of nanostructures in which nonlocal beam, plate, and shell models have been used[11, 12, 13, 14, 15, 16, 17, 18, 19, 20].
The surface stress is another important small scale influence that can significantly affect the mechanical behavior of ultra-thin nanostructures. The concept of surface stress in solids was first proposed by Gibbs[21]. It can be briefly explained as follows. Atoms at or near a free surface of a solid body have different equilibrium requirements in comparison with those within the bulk of material as a consequence of different environmental conditions. Because the energies of atoms located on the surface or near to it are different from those of atoms in the bulk phase, creation of a surface leads to an excess free energy in the solid which is called the surface free energy, and the surface stress is defined based on the variation of surface free energy with the surface strain. In nanostructures for which the surface-to-volume ratio is very high, the surface stress effect plays an important role in their mechanical behavior. In this regard, Cammarata[22] commented that for a solid phase with one or more of its dimensions smaller than about 10 nm, the surface and interface stresses can be principal factors in determining the equilibrium structure and behavior of the solid.
So as to incorporate the surface stress effect into the continuum mechanics, Gurtin and Murdoch[23, 24] developed a non-classical continuum theory. In the Gurtin-Murdoch model, the surface stress was formulated as a function of the deformation gradient, and the surface was treated as a mathematical layer with zero thickness perfectly bonded to the bulk of material. Several research papers on the mechanics of nanorods[25], nanobeams[26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], nanoplates[40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], nanowires[54, 55, 56, 57], and nanoshells[58, 59] can be found, in which the surface stress has been taken into consideration using such a continuum model. Eltaher et al.[27] studied the coupling effects of surface energy and nonlocal elasticity on the vibration behavior of nanobeams based on a finite element approach. In order to capture the surface effects, they developed a sizedependent Euler-Bernoulli beam model based on the Gurtin-Murdoch theory. Amirian et al.[31] investigated the free vibration characteristics of Timoshenko nanobeams made of alumina in the presence of surface and thermal effects resting on a Pasternak foundation. Their results indicated that the surface effects result in an increase in the natural frequency of nanobeams. The asymmetric bifurcation of initially curved Euler-Bernoulli nanobeams with considering surface effects was analyzed by Chen and Meguid[34]. Wang and Wang[36] presented a continuum model for nano-cantilever switches including surface effects and nonlinear curvature. Radebe and Adali[51] studied the influence of surface stress on the buckling response of nonlocal nanoplates subjected to material uncertainty. The surface stress effect on the stiffness of micro- and nano-cantilevers was also probed by Qiao and Zheng[57].
Nanoshells have recently attracted attention of researchers for several applications[60]. In some of these applications, accurate identification of buckling and postbuckling behaviors of the nanoshell can be very significant from the designing point of view. At the macroscale, the postbuckling characteristics of shells have been widely studied[61, 62, 63, 64, 65]. However, research works on the postbuckling behavior of nanoshells with considering the size effects are very limited. Also, a literature survey shows that the surface effects have been extensively investigated on the mechanical behavior of nanobeams, nanoplates, and nanowires. However, the mechanical responses of nanoshells with the consideration of such important effects have been rarely investigated up to now. The limited number of research works on the size-dependent shell models might be because of the difficulty of derivation and solution of non-classical shell equations including small scale effects that are obviously more complex than their beam and plate counterparts.
In the present paper, an analytical size-dependent shell model including the surface effects is proposed to address nonlinear buckling and postbuckling problems of cylindrical nanoshells subjected to the axial compressive load. To this end, within the framework of Gurtin-Murdoch surface elasticity theory, a Donnell-type shell model is developed first. The geometric nonlinearity is taken into account using the von K´arm´an assumption. Then, in the context of variational formulation, an efficient analytical approach is used for the solution procedure. The selected numerical results are presented for investigating the effects of the surface stress, the surface residual tension, and the radius-to-thickness ratio on prebuckling and postbuckling characteristics of nanoshells.
2 Formulation of problemConsider a circular cylindrical nanoshell with the length L, the thickness h, and the midsurface radius R as indicated in Fig. 1. A bulk part and two thin surface layers (inner and outer layers) are considered for the nanoshell. For the bulk part, one has Young’s modulus E and Poisson’s ratio ν. Also, the two surface layers are considered to have the surface Lam´e constants of λs and μs and the surface residual tension τs.
|
| Fig. 1 Schematic of circular cylindrical nanoshell: kinematic parameters, coordinate system, and geometry |
With reference to a coordinate system (x, y, z) with its origin located on the middle surface of nanoshell, the coordinates of a typical point in the axial, circumferential, and radial directions are denoted by x, y, and z, respectively. On the basis of the classical deformation shell theory and Love’s hypotheses, because of assuming zero transverse shear deformation, the rotation of the normal is related to the slope of the mid-surface after deformation. Therefore, the displacement field can be written as follows[66]:
in which u, v, and w denote the middle surface displacements.
Following the classical Donnell shell theory[67] which assumes that the thickness of the shell h is remarkably small as compared with its radius of curvature R, and based on the von K´arm´an hypothesis, the relations between the strain and displacement components are given by
Then, the constitutive relations are expressed as
in which λ = Eν/(1 - ν2) and μ = E/(2(1 + ν)) stand for the classical Lam´e constants.
In nanostructures, there are always interactions between the elastic surface and the bulk of material. Therefore, they mostly undergo in-plane loads in different directions. These inplane loads applied on the surfaces of the bulk part result in surface stresses. According to the Gurtin-Murdoch theory, the surface constitutive equations are given by[23]
where λs and μs are the surface Lam´e constants. Using Eq. (4), the surface stress components
In the classical continuum theories, it is assumed that σzz is equal to zero, because it is small in comparison with the other normal stresses. However, this assumption does not satisfy the surface conditions of the Gurtin-Murdoch theory. In order to solve this problem, it is assumed that σzz varies linearly through the thickness of nanoshell and satisfies the balance conditions on the surfaces. Hence, σzz is obtained as
where the superscripts s+ and s- show the outer and inner layers, respectively.
Using Eq. (5) leads to
In addition, the normal stresses (σxx, σyy) for the bulk of the nanoshell are written as
The total strain energy of the nanoshell including the surface stress effect can be formulated as
where S is the area occupied by the middle plane of the nanoshell. In Eq. (9), the in-plane force resultants, the bending moments, and the shear forces are expressed as
in which
Now from Eq. (2), the compatibility equation is obtained as
Substituting Eq. (13) into Eq. (10) yields
where
By inserting Eq. (14) into Eq. (12), one can obtain
Substituting Eqs. (13) and (14) into Eq. (9) yields
where
It is assumed that the nanoshell is subjected to an axial load. By means of Eqs. (2) and (14), the work done by the external forces is
where σ0x is the average axial stress on the end sections of nanoshell and is positive for the compression.
The total potential energy of the system is
Based on Eqs. (2) and (14),
Similarly, the average end-shortening ratio of the nanoshell can be given as
The deflection of combined axially and laterally loaded shells under simply-supported boundary conditions can be expressed as follows[68]:
in which α = mπ/L, β = n/R, and m and n are the axial half-wave number along the x-axis and the wave number along the y-axis, respectively. Also, f0, f1, and f2 are unknown amplitudes. f0 denotes the uniform deflection in the prebuckling state. f1 sin(αx) sin(βy) expresses the linear buckling shape, and f2(sin(αx))2 represents the nonlinear diamond buckling shape of large deflection.
Substitution of Eq. (23) into Eq. (16) yields
where
Then, the general solution of F is given by
in which σ0y signifies the average circumferential stress and is positive when the nanoshell is circumferentially compressed. The other parameters are introduced as
where
By Eqs. (23) and (26), Eqs. (17) and (19) are rewritten as
By substituting Eqs. (29) and (30) into Eq. (20), the total potential energy is obtained, and the Ritz energy method is then applied as
By Eqs. (23) and (26), Eq. (21) becomes
Through inserting Eq. (32) into Eq. (20), and then from Eq. (31), one has
Considering Eqs. (32) and (33) leads to
As this equation is satisfied, the close condition and 
hold simultaneously. This
indicates that the prebuckling circumferential stress should be related only to the radial pressure
when the shell can freely move in the radial direction. By considering Eq. (34) in the last two
partial derivatives of Eq. (31) and noting that f1 ≠ 0, one has
where
According to Eqs. (36) and (37), one achieves the following relation:
that can be used to obtain the nonlinear critical condition for the axially loaded circular cylindrical nanoshell. By dropping the nonlinear buckling shape (f2 = 0), the linear critical axial load is calculated.
With the use of Eq. (34), substituting Eqs. (23) and (26) into Eq. (22) yields the expression of the end shortening ratio of the nanoshell as
In this section, selected numerical results are presented for nonlinear buckling and postbuckling characteristics of nanoshells under the axial compressive load with and without considering the surface stress effect. The nanoshells are assumed to be made of two materials including Si (100) and Al (111). The material properties of bulk and surface layers of nanoshell used to generate the numerical results are given in Table 1[54, 69, 70]. Also, the dimensions of the nanoshells considered here are selected in accordance with the experimental evidences[60, 71].
As it was mentioned earlier, the nonlinear critical condition for the axially loaded nanoshell can be obtained through Eq. (38). The nonlinear critical condition is defined here as the possible lowest point of external axial load[72]. Figures 2 and 3 show the diagrammatic sketch of solving the nonlinear critical axial stress and the buckling mode of nanoshells made of Si and Al based on the classical and surface elasticity theories. In these figures, for different combinations of mode numbers (m, n), σx0 is plotted against f2/h using Eq. (38). From the lowest point of the envelope curve (indicated by solid lines), the nonlinear critical condition with the nonlinear critical axial stress and the associated nonlinear buckling mode can be extracted.
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| Fig. 2 Diagrammatic sketch of solving nonlinear critical axial stress and buckling mode of nanoshells made of Si and Al based on classical elasticity theory (h = 10 nm, L = R, R = 200h, and n = 4, 5, …, 9) |
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| Fig. 3 Diagrammatic sketch of solving nonlinear critical axial stress and buckling mode of nanoshells made of Si and Al based on surface elasticity theory (h = 50 nm, L = R, and R = 200h) |
In Fig. 4, the postbuckling paths of nanoshells made of Si and Al are shown, in which the solid lines specify the postbuckling equilibrium path. It is seen that the structure follows a prebuckling path before reaching the linear bifurcation point. At this point, the axial stress has its maximum value (the linear critical load or the upper critical load). It is then observed that the structure follows the postbuckling path and the axial stress noticeably decreases up to the minimum value which corresponds to the lower critical load. After that, the stress slightly increases. According to the figure, continuous mode jumps are also observed in the postbuckling regions (m = 1, n = 8, 7, …, 4 in the case of Si, and m = 1, n = 6, 5, …, 3 in the case of Al).
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| Fig. 4 Diagrammatic sketch of load-shortening ratio response curve of nanoshells made of Si and Al based on surface elasticity theory (h = 10 nm, L = R, and R = 200h) |
Figure presents the postbuckling paths of the nanoshells made of Si and Al based on both classical and surface elasticity theories. In this figure, the influence of material type is highlighted. As it can be seen that there is a considerable difference between the prebuckling and postbuckling paths of the nanoshells made of Si and Al when the surface effects are taken into account. It is observed that the curve corresponding to Al is higher than that for Si.
Figure 5 shows the postbuckling paths of nanoshells made of Si with different thicknesses including the surface effects. The results from the classical elasticity theory are also provided in this figure for comparison purpose. It is seen that as the thickness of nanoshell decreases, the surface stress effect becomes more prominent leading to the increase of carrying capacity of nanoshell. Figure 6 indicates that the upper and lower critical loads get larger and the postbuckling equilibrium path moves upward with decreasing the thickness of nanoshell. Moreover, one can find that the difference between the results of two theories is considerable at small values of thickness, but it becomes insignificant by increasing the thickness. It can be explained by the fact that with decreasing the thickness of nanoshell, the energies of surface layers become considerable as compared with those of bulk of material. However, with increasing the thickness of nanoshell, the energies of the bulk phase increase, and thus the surface energies become negligible.
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| Fig. 5 Effect of material type on load-shortening ratio response curve of nanoshells based on classical elasticity theory and surface elasticity theory (L = R, R = 200h, and h = 50 nm) |
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| Fig. 6 Effect of surface stress on load-shortening ratio response curve of nanoshell made of Si (L = R, R = 200h, and n = 4, 5, …, 9) |
Figure 7 highlights the influence of the surface residual tension on the postbuckling paths of nanoshell made of Si. It is observed that the buckling behavior of nanoshell is dependent on the selected value for τs. It is seen that by selecting a positive value for the surface residual tension, the upper and lower critical axial loads increase, and the postbuckling equilibrium path moves upward, whereas the negative value has a reverse effect. The reason for such behaviors is that the tensile and compressive in-plane forces are generated in the nanoshell due to the positive and negative surface residual tensions, respectively.
|
| Fig. 7 Effect of surface residual tension on load-shortening ratio response curve of nanoshell made of Si (h = 60 nm, L = R, R = 200h, and n = 4, 5, …, 9) |
Figure 8 represents the effect of the radius-to-thickness ratio on prebuckling and postbuckling characteristics of the nanoshell. It is observed that by increasing this ratio, the lower critical load decreases, and the postbuckling equilibrium path moves downward.
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| Fig. 8 Effect of radius-to-thickness ratio on load-shortening ratio response curve of nanoshell made of Si based on surface elasticity theory (h = 50 nm and L = R) |
The geometrically nonlinear buckling and postbuckling characteristics of circular cylindrical nanoshells subjected to the axial compressive load including the surface effects are investigated based on an analytical approach. To accomplish this aim, a Donnell-type shell model is first developed, whose geometric nonlinearity is captured based on the von K´arm´an hypothesis. The surface effects are also incorporated into the model according to the Gurtin-Murdoch theory. The selected numerical results are provided on prebuckling and postbuckling characteristics of nanoshells made of Si and Al based on both classical and Gurtin-Murdoch theories. It is shown that there is a considerable difference between the prebuckling and postbuckling paths of the nanoshells made of Si and Al as the surface stress effect is considered. It is also concluded that due to the surface stress effect, the upper and lower critical loads increase, and the postbuckling equilibrium path moves upward when the thickness of nanoshell decreases. The results reveal that the surface energies have significant influence on the postbuckling behavior of very thin nanoshells, but their effect becomes negligible when the thickness is sufficiently large. Another finding is that the surface stress effect is dependent on the value of surface residual tension.
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