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An isogeometric approach of static and free vibration analyses for porous FG nanoplates
European Journal of Mechanics / A Solids 78 (2019) 103851
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An isogeometric approach of static and free vibration analyses for porous FG nanoplates P. Phung-Van a, Chien H. Thai b, c, H. Nguyen-Xuan d, M. Abdel-Wahab b, c, * a
Faculty of Civil Engineering, Ho Chi Minh City University of Technology (HUTECH), Ho Chi Minh City, Viet Nam Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam c Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam d CIRTECH Institute, Ho Chi Minh City University of Technology (HUTECH), Ho Chi Minh City, Viet Nam b
A R T I C L E I N F O
A B S T R A C T
Keywords: Porosities Nonlocal theory Isogeometric analysis (IGA) Porous functionally graded nanoplates Higher order shear deformation theory
This paper presents porosity-dependent analysis of functionally graded nanoplates, which are made of two kinds of porous materials, based on isogeometric approach for the first time. Material properties of the nanoplates are described by using a modified power-law function. The Eringen’s nonlocal elasticity is used to capture the size effects. Using the Hamilton’s principle, the governing equations of the porous FG nanoplates using the higher order shear deformation theory are derived. The obtained results demonstrate the significance effect of nonlocal parameter, material composition, porosity factor, porosity distributions, volume fraction exponent and geometrical parameters on static and free vibration analyses of nanoplates.
1. Introduction Materials with porosities as metal foams are one of the most important categories of lightweight materials. The porous volume fraction usually causes a smooth change in mechanical properties. With a fast growth of nanoscale structures, functionally graded materials (FGMs), as specific nanostructures, are called “new revolutionary of materials”. FGMs are often made of two or more materials such as metal and ceramics, which continuously change through the thickness direc tion and have many advantages because of their constituent phases. For example, with ceramic constituents, they can withstand extreme tem perature environments because of their thermal resistance properties, while stronger mechanical performances are found in metal constitu ents. Therefore, FGMs have been used and applied in several engineer ing fields such as civil constructions, biomedical engineering, automotive industry, nuclear engineering, aerospace engineering, etc. Due to their outstanding mechanical, chemical and electrical prop erties, FGM plates/beams with nano or micro length scale have received the attention of researchers and scientists. As we know, the size effect in nanostructures are absent in the classical continuum theory. Therefore, it overpredicts nanostructures behaviors. To overcome these shortcom ings, Eringen (1972) developed the nonlocal elasticity theory based on the continuum mechanics theory by adding a length scale parameter,
which consists of small size effects with good accuracy to nano structures, into the constitutive equations. Based on classical plate the ory (CPT) and finite element method, free vibration analysis of FG nanoplate was reported in Ref. (Natarajan et al., 2012). Buckling behavior of FG nanobeams was conducted in Ref. (Eltaher et al., 2013). Free vibration analysis of FG nanobeams was also examined by Alshorbagy et al. (2011). A refined four-variable plate model for thermal buckling analysis of FG nanoplates under uniform temperature distri butions was investigated by Barati et al. (Barati and Shahverdi, 2016). Besides, a new four-variable plate theory in vibration analysis of FG nanoplates (Belkorissat et al., 2015) was introduced. A new quasi 3D nonlocal plate theory for vibration and buckling of FGM nanoplates was performed in Ref. (Sobhy and Radwan, 2017). Using Navier solutions, free vibration and buckling analyses of FG nanoplate under thermal load (Ansari et al., 2015) were investigated. A nonlocal beam theory for static, buckling and free vibration analyses (Thai, 2012) was reported. Using higher order shear deformation theory, analytical solutions of post-buckling analysis of FG nanoplates considering porosity distribu tions were reported in Ref. (Barati and Zenkour, 2018). Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams using the generalized differential quadrature method (GDQM) (Shafiei et al., 2017) was introduced. Recently, Phung-Van (Phung-Van et al., 2017a, 2017b, 2018) used isogeometric
* Corresponding author. Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam. E-mail address: [email protected] (M. Abdel-Wahab). https://doi.org/10.1016/j.euromechsol.2019.103851 Received 18 August 2018; Received in revised form 22 August 2019; Accepted 6 September 2019 Available online 17 September 2019 0997-7538/© 2019 Elsevier Masson SAS. All rights reserved.
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European Journal of Mechanics / A Solids 78 (2019) 103851
Fig. 1. Geometry of FGM nanoplates with two porosity distributions.
Fig. 2. Young’s modulus of porous Al/ZrO2-1.
Table 1 Material properties of FGM. E ⱱ
ρ
Al
ZrO2-2
Ti-Al-4V
SUS304
Al2O3
ZrO2-1
Si3N4
70 � 109 0.3 2707
151 � 109 0.3 3000
320.2 � 109 0.26 3750
201.04 � 109 0.3 8166
380 � 109 0.3 3800
200 � 109 0.3 5700
384.43 � 109 0.3 2370
2
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European Journal of Mechanics / A Solids 78 (2019) 103851
Table 2 Non-dimensional deflections of porous Al/Al2O3 nanoplates with ξ ¼ 0. N
0 2 5
Model
IGA-M (Phung-Van et al., 2017b) IGA-P (Phung-Van et al., 2017b) Present IGA-M (Phung-Van et al., 2017b) IGA-P (Phung-Van et al., 2017b) Present IGA-M (Phung-Van et al., 2017b) IGA-P (Phung-Van et al., 2017b) Present
a/h ¼ 5
a/h ¼ 50
μ
μ
0
1
4
0
1
4
0.0903 0.0903 0.0922 0.2721 0.2284 0.2313 0.3246 0.2856 0.2855
0.1060 0.106 0.1095 0.3192 0.2682 0.2748 0.3803 0.3348 0.3389
0.1530 0.1530 0.1614 0.4605 0.3876 0.4051 0.5476 0.4823 0.4991
0.0750 0.0750 0.0750 0.2222 0.1928 0.1923 0.2564 0.2281 0.2279
0.0886 0.0886 0.0888 0.2625 0.2277 0.2278 0.3028 0.2694 0.2701
0.1294 0.1294 0.1302 0.3834 0.3325 0.3346 0.4422 0.3934 0.3967
Table 3 Porous effects on deflections of Al/Al2O3 nanoplates. N
0
ξ
0.1 0.2 0.3 0.4
1
0.1 0.2 0.3
5
0.1 0.2
10
0.1 0.2
Model
PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II
a/h ¼ 5
a/h ¼ 50
μ
μ
0
1
4
0
1
4
0.0997 0.0941 0.1082 0.0962 0.1177 0.0982 0.1285 0.1004 0.2113 0.1884 0.2559 0.1979 0.3251 0.2085 0.3791 0.3121 0.5888 0.3468 0.4318 0.3527 0.6901 0.3968
0.1185 0.1118 0.1286 0.1142 0.1400 0.1167 0.1529 0.1193 0.2512 0.2238 0.3043 0.2351 0.3869 0.2478 0.4503 0.3705 0.7002 0.4116 0.5127 0.4184 0.8209 0.4705
0.1749 0.1649 0.1900 0.1685 0.2069 0.1722 0.2262 0.1760 0.3708 0.3301 0.4495 0.3469 0.5720 0.3656 0.6639 0.5455 1.0344 0.6060 0.7552 0.6155 1.2108 0.6918
0.0812 0.0765 0.0881 0.0780 0.0958 0.0795 0.1045 0.0811 0.1776 0.1571 0.2170 0.1649 0.2793 0.1737 0.3057 0.2470 0.4875 0.2711 0.3368 0.2706 0.5410 0.2957
0.0962 0.0906 0.1043 0.0924 0.1135 0.0942 0.1238 0.0961 0.2105 0.1862 0.2571 0.1954 0.3311 0.2058 0.3624 0.2928 0.5785 0.3214 0.3992 0.3207 0.6417 0.3505
0.1411 0.1328 0.1531 0.1355 0.1665 0.1382 0.1816 0.1410 0.3090 0.2733 0.3776 0.2869 0.4864 0.3021 0.5325 0.4300 0.8512 0.4723 0.5863 0.4709 0.9435 0.5148
approach for size-dependent analysis of functionally graded carbon nanotube reinforced composites nanoplates (Phung-Van et al., 2017a, 2018) and functionally graded nanoplates (Phung-Van et al., 2017b). In their study, materials without voids and porosities have been examined. In the real materials, nevertheless, there are many porosities and voids, which affect stiffness and reliability of structures. This plays an impor tant role in materials design. Therefore, it motivates us to investigate a new class of materials, i.e. porous materials to give a general and whole view in materials science. By using nonlocal strain theory (NSGT), Li et al. (2015) studied wave propagation in FG nanobeams. Buckling analysis of orthotropic nano plates under thermal environments using exact and differential quad rature methods was reported in Ref. (Farajpour et al., 2016). Analytical solutions of vibration analysis of porous functionally graded nanoplates with attached mass (Shahverdi and Barati, 2017) were recently con ducted. Two length scale parameters were considered in their study. Size-dependent analysis of functionally graded isotropic and sandwich microplates using an improved moving Kriging meshfree (Thai et al., 2018a) and IGA (Thai et al., 2018b) was introduced. Nonlinear and transient isogeometric analysis of FG microplate (Thai et al., 2017) was also reported. It seems that only a few papers were published in porous nanoplates, while understanding its mechanics behavior is highly demanded. All previous studies used analytical solutions to examine nanoplates with
porosities. However, analytical methods are limited to simple problems that can be modelled by partial differential equations. Hence, this paper aims to fill in this gap and makes the first attempt to calculate porous functionally graded nanoplate using numerical methods. The proposed model based on isogeometric analysis and nonlocal continuum theory is developed. Using the nonlocal theory of Eringen, the porousdependency, size-dependency and length scale effect of the nanoplate are performed. Two types of porosity-dependent material properties are incorporated to the modified power-law modeling. The results indicate that behaviors of the nanoplate are significantly influenced by the nonlocal parameter, geometrical parameters and material composition. The obtained results can be considered as benchmark results to analyze porous FG nanoplates. 2. Theoretical formulation 2.1. Porous functionally graded materials We consider a porous nanoplate (length a, width b and thickness h) made of metal and ceramic as indicated in Fig. 1. Two porosity distri butions through thickness of the nanoplate including even porosities (PFGM-I) and uneven porosities (PFGM-II) have been considered as illustrated in Fig. 1. For PFGM-I, porosities are randomly distributed in the cross-section of the nanoplate. Moreover, porosities in PFGM-II are 3
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European Journal of Mechanics / A Solids 78 (2019) 103851
Fig. 3. Young’s modulus of porous Al/Al2O3.
Fig. 5. Deflections of an Al/Al2O3 nanoplate with μ ¼ 1 and n ¼ 1.
Fig. 4. Deflections of an Al/Al2O3 nanoplate with μ ¼ 4 and a/h ¼ 5.
4
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Table 4 Porous effects on deflections of Al/ZrO2-2 nanoplates with a/h ¼ 10. N
1
ξ
0.1 0.2 0.3 0.4 0.5 0.6
3
0.1 0.2 0.3 0.4 0.5
Model
PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II
Vm þ Vc ¼ 1
;
Vc ¼
μ 0
1
2
3
4
0.3354 0.3036 0.3897 0.3149 0.4623 0.3271 0.5659 0.3402 0.7293 0.3545 1.0394 0.3701 0.3888 0.3473 0.4634 0.3624 0.5709 0.3791 0.7443 0.3976 1.0920 0.4184
0.3986 0.3608 0.4633 0.3743 0.5497 0.3888 0.6729 0.4044 0.8673 0.4214 1.2362 0.4400 0.4621 0.4127 0.5509 0.4308 0.6789 0.4506 0.8853 0.4727 1.2992 0.4975
0.4619 0.4179 0.5368 0.4336 0.6370 0.4505 0.7799 0.4686 1.0054 0.4883 1.4330 0.5099 0.5355 0.4782 0.6385 0.4991 0.7869 0.5222 1.0264 0.5477 1.5064 0.5766
0.5251 0.4751 0.6104 0.4930 0.7243 0.5121 0.8869 0.5328 1.1434 0.5552 1.6298 0.5798 0.6088 0.5436 0.7260 0.5675 0.8949 0.5937 1.1674 0.6228 1.7136 0.6557
0.5883 0.5323 0.6839 0.5523 0.8117 0.5738 0.9939 0.5970 1.2814 0.6221 1.8267 0.6497 0.6822 0.6091 0.8136 0.6359 1.0030 0.6653 1.3084 0.6979 1.9208 0.7348
�n � 1 z þ 2 h
(2)
in which subscript c and m represent ceramic and metal, respectively and n is volume fraction exponent. According to the multi-step sequential infiltration technique, all material properties of PFGM such as density, Poisson’s ratio, Young’s modulus, etc., are defined as: PðzÞ ¼ ðPc
Pm ÞVc þ Pm
PðzÞ ¼ ðPc
Pm ÞVc þ Pm
ξ for PFGM I 2 � � ξ 2jzj 1 for PFGM ðPc þ Pm Þ 2 h ðPc þ Pm Þ
(3) II
From Eq. (3), when ξ is zero, PFGM becomes perfect FGM. Moreover, the expressions of Young’s modulus of PFGM can be formulated as: EðzÞ ¼ ðEc
Em ÞVc þ Em
EðzÞ ¼ ðEc
Em ÞVc þ Em
ξ for PFGM I 2 � � ξ 2jzj 1 for PFGM ðEc þ Em Þ 2 h ðEc þ Em Þ
(4) II
Young’s modulus distributions of Al/ZrO2-1 through the thickness of the nanoplate are also indicated in Fig. 2. We can observe that Young’s modulus without the porosities (ξ ¼ 0) indicated in Fig. 2a is continuous through the top surface (ceramic rich) and the bottom surface (metal rich). Porous effect on Young’s modulus is also performed in Fig. 2b and c. For PFGM-I, forms of curves of Young’s modulus shown in Fig. 2b are the same as those in Fig. 2a with a decrease in Young’s modulus amplitude, leading to decrease in the stiffness of the plate as well. For PFGM-II, the porosities are distributed around the middle zone and tend to zero at the top and the bottom of the cross-section. Therefore, we can see that Young’s modulus shown in Fig. 2c is maximum at the top and the bottom and decreases towards middle zone direction, as clearly shown in Fig. 2d. At the top and the bottom, there is no porosity reached,
distributed around middle zone of the cross-section. Based on the modified rule of mixture, the material properties of porous functionally graded materials (PFGM) are expressed as: � � ξ� ξ� PðzÞ ¼ Pc Vc þ Pm Vm (1) 2 2 where ξ is porosity volume fraction, Vc and Vm are volume fractions of ceramic and metal defined as:
Table 5 The first two non-dimensional natural frequencies of a SSSS SUS304/Si3N4 square nanoplates with a ¼ 10, a/h ¼ 10 and ξ ¼ 0. n
Model
Mode 1
Mode 2
μ
2 10
Ref. (Phung-Van et al., 2017a) Present Ref. (Phung-Van et al., 2017a) Present
μ
0
1
2
4
0
1
2
4
0.0485 0.0466 0.0416 0.0400
0.0443 0.0426 0.0380 0.0365
0.0410 0.0395 0.0352 0.0338
0.0362 0.0349 0.0311 0.0299
0.1154 0.1138 0.0990 0.0975
0.0944 0.0930 0.0810 0.0797
0.0819 0.0806 0.0702 0.0691
0.0669 0.0659 0.0574 0.0564
Fig. 6. Young’s modulus of porous Al/ZrO2-2. 5
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European Journal of Mechanics / A Solids 78 (2019) 103851
Fig. 9. Deflections of a Ti-Al-4V/Al2O3 nanoplate with n ¼ 1.5, a/h ¼ 20 and μ ¼ 2.
Fig. 7. Deflections of an Al/ZrO2-2 nanoplate (PFGM-I) with n ¼ 1 and a/h ¼ 10.
where αðjx xj; τÞ is the nonlocal kernel function; σij is the local stress tensor at point x, and τ is a length scale parameter. According to Hooke’s law, the local stress is also defined as: 0
(7)
σ ij ¼ Cijkl εkl
where Cijkl is the elasticity tensor. Following Ref. (Eringen, 1972), Eq. (6) can be rewritten as: � 1 μr2 tij ¼ σ ij (8) where μ is a nonlocal parameter and r2 ¼ ∂2 =∂x2 þ ∂2 =∂y2 is the Laplace operator. The governing equations for non-local elastic can be given as: � � σ ij;j þ 1 μr2 fi ¼ 1 μr2 ρu€i (9) To transform a strong form to a weak form, we multiply Eq. (9) with δui and then performing integration over the domain together with a chain of integration, then we obtain the following form: Z Z Z Z � � 1 μr2 ρu€i δui dV ¼ 1 μr2 fi δui dV þ σij nj δui dΓ σ ij δεij dV þ V
Fig. 8. Deflections of an Al/ZrO2-2 nanoplate (PFGM-II) with n ¼ 1 and a/h ¼ 10.
V
Γt
(10)
2.3. Higher order shear deformation theory
i.e. Young’s modulus amplitude of PFGM-II is that of FGM. Moreover, Young’s modulus amplitude of PFGM-II equals to that of PFGM-I at the mid surface.
In equivalent single-layer theories, two theories including classical laminated theory (CLT) and the first-order shear deformation theory (FSDT) are commonly used. However, solutions achieved by CLT and FSDT are not correct because the interlaminar shear deformation in CLT is neglected, while the shear correction factor is required to ensure the stability of the solution of FSDT. To overcome those difficulties, in this study, the higher-order shear deformation theory (HSDT) without the shear correction factor is used. In HSDT, the transverse shear stresses are represented correctly. The displacement field of the porous FGM nano plate based on HSDT can be formulated as follows:
2.2. Nonlocal continuum theory The equations of nonlocal elastic solids are considered herein and formulated as: tij;j þ fi ¼ ρu€i
V
(5)
where i,j represent the symbols z, y, x; fi is the body load and ρ is the density. Based on the nonlocal continuum theory of Eringen (1972), the stress tensor can be expressed as follows: Z 0 0 0 tij ¼ αðjx xj; τÞσij ðx Þdx (6)
uðx; y; zÞ ¼ u0 ðx; yÞ vðx; y; zÞ ¼ v0 ðx; yÞ wðx; y; zÞ ¼ w0 ðx; yÞ
V
6
∂w þ f ðzÞβx ðx; yÞ ∂x ∂w z þ f ðzÞβy ðx; yÞ ∂y z
(11)
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European Journal of Mechanics / A Solids 78 (2019) 103851
Fig. 10. Porous effects on deflections of a FGM nanoplate with n ¼ 1.5, a/h ¼ 20 and μ ¼ 2.
Fig. 11. Young’s modulus of PFGM with n ¼ 1.5, ξ ¼ 0:3 and μ ¼ 2.
Table 6 The six lowest frequencies of a SSSS Al/Al2O3 with n ¼ 3. μ
ξ
Type
0
0 0.1
FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II
0.2 0.3 1
0 0.1 0.2 0.3
2
0 0.1 0.2 0.3
Mode 1
2
3
4
5
6
0.0599 0.0554 0.0592 0.0485 0.0584 0.0346 0.0572 0.0547 0.0506 0.0541 0.0443 0.0533 0.0316 0.0523 0.0507 0.0469 0.0501 0.0411 0.0494 0.0293 0.0484
0.1456 0.1350 0.1437 0.1183 0.1414 0.0845 0.1382 0.1191 0.1104 0.1176 0.0968 0.1156 0.0691 0.1130 0.1032 0.0957 0.1019 0.0839 0.1002 0.0599 0.0980
0.1456 0.1350 0.1438 0.1183 0.1414 0.0845 0.1382 0.1191 0.1104 0.1176 0.0968 0.1157 0.0691 0.1130 0.1032 0.0957 0.1019 0.0839 0.1002 0.0599 0.0980
0.2127 0.2060 0.2099 0.1823 0.2062 0.1311 0.2013 0.1668 0.1548 0.1646 0.1361 0.1618 0.0980 0.1581 0.1388 0.1289 0.1370 0.1134 0.1347 0.0816 0.1316
0.2128 0.2061 0.2099 0.1939 0.2062 0.1619 0.2013 0.1944 0.1809 0.1916 0.1594 0.1879 0.1148 0.1831 0.1589 0.1478 0.1566 0.1302 0.1536 0.0939 0.1496
0.2234 0.2075 0.2205 0.1939 0.2168 0.1628 0.2118 0.1955 0.1820 0.1928 0.1604 0.1892 0.1155 0.1844 0.1598 0.1487 0.1575 0.1310 0.1546 0.0944 0.1507
7
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Table 7 The six lowest frequencies of a CCCC Al/Al2O3 with n ¼ 3. μ
ξ
Type
0
0 0.1
FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II
0.2 0.3 1
0 0.1 0.2 0.3
2
0 0.1 0.2 0.3
Mode 1
2
3
4
5
6
0.1091 0.1014 0.1078 0.0891 0.1061 0.0635 0.1037 0.0981 0.0911 0.0969 0.0799 0.0953 0.0569 0.0932 0.0898 0.0833 0.0887 0.0731 0.0872 0.0519 0.0853
0.2091 0.1950 0.2062 0.1722 0.2025 0.1243 0.1974 0.1671 0.1556 0.1648 0.1372 0.1617 0.0986 0.1577 0.1430 0.1332 0.1410 0.1173 0.1385 0.0841 0.1350
0.2092 0.1951 0.2064 0.1723 0.2026 0.1244 0.1976 0.1672 0.1557 0.1649 0.1373 0.1619 0.0986 0.1579 0.1431 0.1333 0.1412 0.1173 0.1386 0.0841 0.1351
0.3013 0.2811 0.2971 0.2482 0.2916 0.1791 0.2842 0.2194 0.2043 0.2163 0.1800 0.2122 0.1290 0.2068 0.1808 0.1683 0.1782 0.1481 0.1749 0.1059 0.1704
0.3446 0.3227 0.3393 0.2870 0.3324 0.2108 0.3232 0.2380 0.2226 0.2343 0.1975 0.2295 0.1440 0.2232 0.1927 0.1801 0.1897 0.1597 0.1858 0.1162 0.1807
0.3479 0.3256 0.3426 0.2895 0.3357 0.2123 0.3265 0.2412 0.2254 0.2374 0.1999 0.2326 0.1456 0.2262 0.1958 0.1830 0.1928 0.1621 0.1889 0.1179 0.1837
Table 8 The first natural frequency of a CCCC Al/ZrO2-1 with ξ ¼ 0:3. n
BCs
Type
1
SSSS
PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II
CCCC 5
SSSS CCCC
15
SSSS CCCC
20
SSSS CCCC
μ 0
0.5
1.5
2.5
3
3.5
4
0.0764 0.0837 0.0837 0.1536 0.0755 0.0855 0.1389 0.1545 0.0754 0.0852 0.1388 0.1538 0.0745 0.0845 0.1373 0.1527
0.0728 0.0798 0.0798 0.1452 0.0720 0.0815 0.1314 0.1462 0.0720 0.0812 0.1312 0.1455 0.0711 0.0806 0.1298 0.1445
0.0671 0.0735 0.0735 0.1318 0.0663 0.0750 0.1193 0.1328 0.0662 0.0748 0.1192 0.1323 0.0654 0.0742 0.1179 0.1313
0.0625 0.0685 0.0685 0.1215 0.0618 0.0699 0.1100 0.1225 0.0617 0.0697 0.1099 0.1220 0.0609 0.0691 0.1087 0.1212
0.0605 0.0663 0.0663 0.1172 0.0598 0.0677 0.1060 0.1182 0.0598 0.0675 0.1060 0.1177 0.0590 0.0669 0.1048 0.1169
0.0587 0.0643 0.0643 0.1133 0.0580 0.0657 0.1025 0.1143 0.0580 0.0655 0.1025 0.1138 0.0573 0.0649 0.1014 0.1130
0.0571 0.0625 0.0625 0.1097 0.0564 0.0638 0.0993 0.1107 0.0564 0.0636 0.0993 0.1102 0.0557 0.0631 0.0982 0.1095
where u0, v0 and w0 are the displacements in plane and deflection in the z direction; βx and βy are rotations of the plate and f(z) depends on the
2
�
4 3 plate thickness. In this study, f(z) is chosen as fðzÞ ¼ z z . 3h2 Based on the generalize shear deformation theory, the strains can be expressed as: 3 2 ∂u 6 ∂x 7 7 2 3 6 6 7 εxx 6 ∂v 7 4 5 6 7 ε ¼ εyy ¼ 6 7 6 ∂y 7 εxy 6 7 4 ∂u ∂v 5 þ ∂y ∂x 3 2 2 3 2 3 ∂2 w ∂βx ∂u0 6 2 7 6 7 6 7 6 ∂x 7 ∂x ∂x 6 7 6 7 6 7 6 7 6 ∂v 7 6 ∂2 w 7 ∂ β 6 7 y 6 7 6 7 0 6 7 ¼6 þ f ðzÞ6 (12) 7 þ z6 2 7 7 ∂ y 6 7 6 ∂x ∂ y 7 7 6 6 7 6 7 7 6 4 ∂u0 ∂v0 5 6 7 4 ∂βx ∂βy 5 4 ∂2 w 5 þ þ 2 ∂y ∂x ∂y ∂x ∂x∂y
γ¼
γ xz γ yz
�
3
∂u ∂w � � 6 ∂z þ ∂x 7 β 0 6 7 ¼6 7 ¼ f ðzÞ βx 4 ∂v ∂w 5 y þ ∂z ∂y
(13)
In a shorter form, Eq. (12) can be rewritten as:
ε ¼ εm þ zκ1 þ f ðzÞκ2 ;
0
γ ¼ f ðzÞεs
(14)
where 2
3 3 3 2 2 � � βx;x u0;x w0;xx β εm ¼ 4 v0;y 5 ; κ1 ¼ 4 w0;yy 5; κ2 ¼ 4 βy;y 5 ; εs ¼ x βy βx;y þ βy;x u0;y þ v0;x 2w0;xy
(15)
According to Hooke’s law, the stresses are defined as: �T σ ¼ σ xx σ yy σ xy ¼ Cb ε ¼ Cb ðεm þ zκ1 þ f ðzÞκ2 Þ �T 0 τ ¼ τxz τyz ¼ Cs γ ¼ Cs f ðzÞεs Based on Eq. (8), the stress resultants are formulated as:
8
(16)
P. Phung-Van et al.
European Journal of Mechanics / A Solids 78 (2019) 103851
Fig. 12. The lowest six mode shapes of a square porous nanoplate.
8 9 8 8 9 9 8 9 Zt=2 < σ xx = Zt=2 < σ xx = �< Nxx = �< Mxx = 2 Nyy ¼ σ yy dz ; 1 μr Myy ¼ σ yy zdz; 1 μr : ; : : ; ; : ; Nxy σ xy Mxy σ xy t=2 t=2 8 9 8 9 t=2 t=2 � � � � Z Z < σxx = �< Pxx = � Qxz τxz 0 σyy f ðzÞdz ; 1 μr2 1 μr2 Pyy ¼ ¼ f ðzÞdz Qyz τyz : ; : ; Pxy σxy t=2 t=2
2
2
Cb ¼
(17) Substituting Eq. (16) 2 3 2 N A �6 7 6 B 2 6M7 6 1 μr 4 5 ¼ 4 P E Q 0
into Eq. (17), we obtain: 32 3 εm B E 0 6 7 D F 0 7 76 κ1 7 F G 0 54 κ2 5 0 0 As εs
ðA; B; D; E; F; GÞ ¼ Cb Z 0 ¼ Cs ½f ðzÞ�2 dz
6 6 νðzÞ 6 νðzÞ 6 4 0
EðzÞ
νðzÞ
0
1
0
2
1
0
3
7 7 7 7 νðzÞ 5
1
;
Cs ¼
� EðzÞ 1 2ð1 þ νðzÞÞ 0
2
0 1
�
(20)
A weak form of the PFGM nanoplate under transverse load q0 can be expressed as: Z Z � � � € ðδεÞT Dmb ε þ ðδγÞT As γ dΩ þ δuT m 1 μr2 udΩ Ω Ω Z � ¼ δ w μr2 w q0 dΩ (21)
(18)
Ω
where Z
1
where � 1; z; z2 ; f ðzÞ; zf ðzÞ; f 2 ðzÞ dz ;
2
3 A B E 6 7 Dmb ¼ 4 B D F 5; E F H Z ðI1 ; I2 ; I3 ; I4 ; I5 ; I6 Þ ¼
As (19)
in which
2
2 I1 I2 7 6 I 0 5; I ¼ 4 I2 I3 0 I I4 I5 h=2 � ρ 1; z; z2 ; f ðzÞ; zf ðzÞ; f 2 ðzÞ dz I 6 m ¼ 40 0
h=2
and
9
0
0
3
I4
3
7 I5 5 I6
(22)
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European Journal of Mechanics / A Solids 78 (2019) 103851
3 3 2 2 v0 u0 3 2 3 u1 w0 7 6 6 ∂w 7 7 6 ∂w 7 6 ; u ; u u ¼ 4 u2 5; u1 ¼ 6 ¼ ¼ 7 2 6 7 3 4 0 5 4 ∂x 5 4 ∂y 5 0 u3 βx βy
From Eq. (29), we observe that the basis functions are required at least second and third order derivatives. Hence, IGA can be considered as the most suitable and strong candidate in order to analyze porous nanoplates.
2
(23)
Note that, we can observe that Eq. (21) is in the third order derivative form. Therefore, the order of the basic functions of approximated dis placements is at least a third order. Therefore, isogeometric analysis (IGA) (Cottrell et al., 2009) is a strong and suitable candidate to analyze nanoplates, which due to IGA can easily achieve any desired degree of smoothness through the choice of the interpolation order. Nowadays, IGA has been successfully used to calculate several fields such as com posite plates/shells (Nguyen et al., 2014, 2017; Phung-Van et al., 2015a, 2015b; Tran et al., 2017; Nguyen-Thanh et al., 2015; Thai et al., 2019a, 2019b), large deformation (Nguyen-Thanh et al., 2014, 2017; Vu-Bac et al., 2018), phase-field analysis (Areias et al., 2016), optimization (Lieu and Lee, 2017) and so on.
4. Numerical results
3. Porous functionally graded nanoplate formulation
w¼
Several examples reported in this paper aim to show our new con tributions in functionally graded nanoplates with porosity. Table 1 lists the material properties of functionally graded materials. 4.1. Static analysis We now consider a square nanoplate made of Al/Al2O3 with po rosities. To verify the present results, porous parameter is set zero. Nondimensional deflection is used and defined as follows:
m�n X
(24)
RI ðξ; ηÞdI
I¼1
where dI ¼ ½u0I v0I wI βxI βyI �T is a vector of degrees of freedom of the nanoplate, and RI is the NURBS basis function (Cottrell et al., 2009). Substituting Eq. (24) into Eqs. (14) and (15), the strains are rewritten as:
εm ¼
m�n X
BmI dI ; κ1 ¼
I¼1
m�n X
Bb1 I dI ; κ2 ¼
I¼1
where 2
RI;x 0 6 ¼ 4 0 RI;y RI;y RI;x � 0 0 0 RI BsI ¼ 00 0 0
BmI
3
Bb2 I dI ; εs ¼
I¼1
2 00 7 6 b1 0 0 0 5 ; BI ¼ 4 0 0 00 000 � 0 000
m�n X
m�n X
BsI dI
(25)
I¼1
3 2 3 RI;xx 0 0 0 0 0 RI;x 0 7 7 6 b2 RI;yy 0 0 5 ; BI ¼ 4 0 0 0 0 RI;y 5; 2RI;xy 0 0 0 0 0 RI;y RI;x
RI
(26)
The governing equations for static and free vibration analyses are formulated by: (27)
Kd ¼ F K
�
(28)
ω2 M d ¼ 0
where Z h i � m b1 b2 � � �T B B B Dmb Bm Bb1 Bb2 þ ðBs ÞT As Bs dΩ K¼ ZΩ � ~ μr 2 R ~Tm R ~ dΩ M¼ R ZΩ � R μr2 R q0 dΩ F¼
; Dm ¼
100Em h3 � 12 1 ν2m
(31)
Effect of nonlocal parameters on non-dimensional deflection of the nanoplate is listed in Table 2. Reference solutions were reported in Ref. (Phung-Van et al., 2017b), and the material properties of func tionally graded materials are based on Mori Tanaka (IGA-M) and the mixture rule or power-law (IGA-P). It can be observed that the present results match well with the reference solutions (Phung-Van et al., 2017b) for all cases. Next, porous effects on deflections of Al/Al2O3 nanoplates are listed in Table 3. It can be seen that with an increase of porous parameter, deflections of the nanoplate increase as well. This is because the increase of porous factor leads to a decrease of stiffness of the nanoplate. Besides, effects of porosities on Young’s modulus of Al/Al2O3 are also investi gated in Fig. 3. We can see that with ξ ¼ 0:4 Young’s modulus is nega tive. Therefore, porous factor in this case is only set lower than 0.4. Volume fraction exponent effects on non-dimensional deflection of the PFGM nanoplate with μ ¼ 4 and a/h ¼ 5 are studied and performed in Fig. 4. Again, it can be seen that the deflection of the plate increases with an increase of the volume fraction exponent. Fig. 5 shows the effects of length-thickness ratios (a/h) on deflection of the plate. We observe that deflections decrease with an increase of a/h. Furthermore, porous effects on deflections of Al/ZrO2-2 nanoplates with a/h ¼ 10 are investigated and listed in Table 4. And porous parameter effects on Young’s modulus are examined in Fig. 6 which shows that a porous factor less than 0.6 leads to positive Young’s modulus. Fig. 7 and Fig. 8 show the deflection against nonlocal factor for PFGM-I and PFGM-II nanoplates, respectively. Again, with an increase of porous parameter, deflection increases, as expected. Furthermore, deflections of a Ti-Al-4V/Al2O3 nanoplate with n ¼ 1.5, a/h ¼ 20 and μ ¼ 2 are drawn in Fig. 9. It can be seen that deflection of PFGM-I is larger than that of PFGM-II and FGM. A com parison between deflections of Al/Al2O3, Al/ZrO2-1 and Al/ZrO2-2 is illustrated in Fig. 10. Distributions of Young’s modulus are also shown in Fig. 11. We can see that deflection is the smallest for Al/Al2O3 and the largest for Al/ZrO2-2. This means that the stiffness of Al/Al2O3 plate is the largest and that of Al/ZrO2-2 is the smallest.
Based on NURBS basis functions (Cottrell et al., 2009), the displacement field is defined as follows: uh ðξ; ηÞ ¼
wDm q0 a4
(29)
Ω
with 8 9 2 RI 0 < R1 = ~ R ¼ R2 ; R1 ¼ 4 0 0 : ; 0 0 R3 2 0 0 RI ¼ 40 0 0 0 0 0
4.2. Free vibration 2 3 0 0 0 0 RI 5 RI;x 0 0 ; R2 ¼ 4 0 0 0 0 0 RI 0 3 0 0 0 05 0 0
3 0 0 0 RI;y 0 0 5; R3 0 0 RI
Firstly, a square nanoplate made of SUS304/Si3N4 is investigated. The non-dimensional frequency is used and defined as: rffiffiffiffiffi ρc Ec ; Gc ¼ ω ¼ ωh (32) Gc 2ð1 þ νc Þ The first two frequencies of the nanoplate with a ¼ 10, a/h ¼ 10 and ξ ¼ 0 are listed in Table 5. We can see that the present results match well with those reported in Ref. (Phung-Van et al., 2017a).
(30)
10
European Journal of Mechanics / A Solids 78 (2019) 103851
P. Phung-Van et al.
Next, effects of nonlocal and porous parameters on the six lowest frequencies of an Al/Al2O3 nanoplate with simply supported (SSSS) and clamped (CCCC) boundary conditions are presented in Table 6 and Table 7, respectively. It can be observed that an increase of porous parameter leads to a decrease of the stiffness of the plate and the fre quencies increase. Besides, the frequencies decrease with an increase of nonlocal parameter was also reported in Ref. (Phung-Van et al., 2017a). Also, the frequencies of the CCCC nanoplate are higher than those of the SSSS nanoplate. Finally, the effect of the nonlocal parameter on the first natural frequency of the porous nanoplate is shown in Table 8. The frequencies of the nanoplate decrease with an increase of the nonlocal parameter, as expected. The first six mode shapes of the porous nanoplate are also plotted in Fig. 12.
Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y., 2009. Isogeometric Analysis, towards Integration of CAD and FEA. Wiley. Eltaher, M.A., Emam, S.A., Mahmoud, F.F., 2013. Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct. 96, 82–88. Eringen, A.C., 1972. Nonlocal polar elastic continua. Int. J. Eng. Sci. 10 (1), 1–7. Farajpour, A., Yazdi, M.R.H., Rastgoo, A., Mohammadi, M., 2016. A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment. Acta Mech. 227 (7), 1849–1867. Li, L., Hu, Y.J., Ling, L., 2015. Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory. Compos. Struct. 133, 1079–1092. Lieu, Q.X., Lee, J., 2017. A multi-resolution approach for multi-material topology optimization based on isogeometric analysis. Comput. Methods Appl. Math. 323, 272–302. Natarajan, S., Chakraborty, S., Thangavel, M., Bordas, S., Rabczuk, T., 2012. Sizedependent free flexural vibration behavior of functionally graded nanoplates. Comput. Mater. Sci. 65, 74–80. Nguyen, V.P., Kerfriden, P., Bordas, S.P.A., Rabczuk, T., 2014. Isogeometric analysis suitable trivariate NURBS representation of composite panels with a new offset algorithm. Comput. Aided Des. 55, 49–63. Nguyen, T.N., Ngo, T.D., Nguyen-Xuan, H., 2017. A novel three-variable shear deformation plate formulation: theory and Isogeometric implementation. Comput. Methods Appl. Mech. Eng. 326, 376–401. Nguyen-Thanh, N., Muthu, J., Zhuang, X., Rabczuk, T., 2014. An adaptive threedimensional RHT-splines formulation in linear elasto-statics and elasto-dynamics. Comput. Mech. 53 (2), 369–385. Nguyen-Thanh, N., Valizadeh, N., Nguyen, M., Nguyen-Xuan, H., Zhuang, X., Areias, P., et al., 2015. An extended isogeometric thin shell analysis based on Kirchhoff–Love theory. Comput. Methods Appl. Mech. Eng. Fract. Mech. 284, 265–291. Nguyen-Thanh, N., Zhou, K., Zhuang, X., Areias, P., Nguyen-Xuan, H., Bazilevs, Y., et al., 2017. Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling. Comput. Methods Appl. Mech. Eng. 316, 1157–1178. Phung-Van, P., De Lorenzis, L., Thai, C.H., Abdel-Wahab, M., Nguyen-Xuan, H., 2015. Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higher-order shear deformation theory and isogeometric finite elements. Comput. Mater. Sci. 96, 495–505. Phung-Van, P., Nguyen, L.B., Tran, L.V., Dinh, T.D., Thai, C.H., Bordas, S.P.A., et al., 2015. An efficient computational approach for control of nonlinear transient responses of smart piezoelectric composite plates. Int. J. Nonlin. Mech. 76, 190–202. Phung-Van, P., Lieu, Q.X., Nguyen-Xuan, H., Wahab, M.A., 2017. Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates. Compos. Struct. 166, 120–135. Phung-Van, P., Ferreira, A.J.M., Nguyen-Xuan, H., Abdel-Wahab, M., 2017. An isogeometric approach for size-dependent geometrically nonlinear transient analysis of functionally graded nanoplates. Compos. Part B Eng. (118), 125–134. Phung-Van, P., Thanh, C.-L., Nguyen-Xuan, H., Abdel-Wahab, M., 2018. Nonlinear transient isogeometric analysis of FG-CNTRC nanoplates in thermal environments. Compos. Struct. 201, 882–892. Shafiei, N., Mirjavadi, S.S., MohaselAfshari, B., Rabby, S., Kazemi, M., 2017. Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Comput. Methods Appl. Math. 322, 615–632. Shahverdi, H., Barati, M.R., 2017. Vibration analysis of porous functionally graded nanoplates. Int. J. Eng. Sci. 120, 82–99. Sobhy, M., Radwan, A.F., 2017. A new quasi 3D nonlocal plate theory for vibration and buckling of FGM nanoplates. Int. J. Appl. Mech. 9 (1). Thai, H.T., 2012. A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 52, 56–64. Thai, S., Thai, H.T., Vo, T.P., Nguyen-Xuan, H., 2017. Nonlinear static and transient isogeometric analysis of functionally graded microplates based on the modified strain gradient theory. Eng. Struct. 153, 598–612. Thai, C.H., Ferreira, A.J.M., Lee, J., Nguyen-Xuan, H., 2018. An efficient size-dependent computational approach for functionally graded isotropic and sandwich microplates based on modified couple stress theory and moving Kriging-based meshfree method. Int. J. Mech. Sci. 142, 322–338. Thai, C.H., Ferreira, A.J.M., Nguyen-Xuan, H., 2018. Isogeometric analysis of sizedependent isotropic and sandwich functionally graded microplates based on modified strain gradient elasticity theory. Compos. Struct. 192, 274–288. Thai, C.H., Ferreira, A., Tran, T., Phung-Van, P., 2019. Free vibration, buckling and bending analyses of multilayer functionally graded graphene nanoplatelets reinforced composite plates using the NURBS formulation. Compos. Struct. 220, 749–759. Thai, C.H., Ferreira, A., Phung-Van, P., 2019. Size dependent free vibration analysis of multilayer functionally graded GPLRC microplates based on modified strain gradient theory. Compos. Part B Eng. 169, 174–188. Tran, L.V., Wahab, M.A., Kim, S.E., 2017. An isogeometric finite element approach for thermal bending and buckling analyses of laminated composite plates. Compos. Struct. 179, 35–49. Vu-Bac, N., Duong, T.X., Lahmer, T., Zhuang, X., Sauer, R.A., Park, H., et al., 2018. A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Comput. Methods Appl. Mech. Eng. 331, 427–455.
5. Conclusions We have for the first time presented static and free vibration analyses of nonlocal porous FG nanoplates using higher order shear deformation theory (HSDT). Material properties of the nanoplates are described via a modified power-law function. The Eringen’s nonlocal elasticity is used to analyze the length scale effects. Based on the Hamilton’s principle, the governing equations of the porous FG nanoplates using the gener alized higher order shear deformation theory are obtained. Based on the present formulations numerical results, the following conclusions are drawn: i. Isogeometric analysis of porosity-dependent FG nanoplates using IGA based on HSDT was presented. ii. The integration between the nonlocal theory and IGA is a suit able, strong and effective candidate to study the porous FG nanoplates. iii. Deflection of the plate increases with the increase of the volume fraction exponent. iv. Deflection of PFGM-I is larger than that of PFGM-II with the in crease of porous parameter. v. For free vibration analysis, the increase of porous parameter leads to the decrease of the stiffness of the plate and increase of the frequencies. Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02–2019.09. References Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F.F., 2011. Free vibration characteristics of a functionally graded beam by finite element method. Appl. Math. Model. 35 (1), 412–425. Ansari, R., Ashrafi, M., Pourashraf, T., Sahmani, S., 2015. Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory. Acta Astronaut. 109, 42–51. Areias, P., Rabczuk, T., Msekh, M., 2016. Phase-field analysis of finite-strain plates and shells including element subdivision. Comput. Methods Appl. Mech. Eng. 312, 322–350. Barati, M.R., Shahverdi, H., 2016. A four-variable plate theory for thermal vibration of embedded FG nanoplates under non-uniform temperature distributions with different boundary conditions. Struct. Eng. Mech. 60 (4), 707–727. Barati, M.R., Zenkour, A.M., 2018. Analysis of Postbuckling Behavior of General HigherOrder Functionally Graded Nanoplates with Geometrical Imperfection Considering Porosity Distributions. Mechanics of Advanced Materials and Structures. Belkorissat, I., Houari, M.S.A., Tounsi, A., Bedia, E.A.A., Mahmoud, S.R., 2015. On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model. Steel Compos. Struct. 18 (4), 1063–1081.
11
Contents lists available at ScienceDirect
European Journal of Mechanics / A Solids journal homepage: http://www.elsevier.com/locate/ejmsol
An isogeometric approach of static and free vibration analyses for porous FG nanoplates P. Phung-Van a, Chien H. Thai b, c, H. Nguyen-Xuan d, M. Abdel-Wahab b, c, * a
Faculty of Civil Engineering, Ho Chi Minh City University of Technology (HUTECH), Ho Chi Minh City, Viet Nam Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam c Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam d CIRTECH Institute, Ho Chi Minh City University of Technology (HUTECH), Ho Chi Minh City, Viet Nam b
A R T I C L E I N F O
A B S T R A C T
Keywords: Porosities Nonlocal theory Isogeometric analysis (IGA) Porous functionally graded nanoplates Higher order shear deformation theory
This paper presents porosity-dependent analysis of functionally graded nanoplates, which are made of two kinds of porous materials, based on isogeometric approach for the first time. Material properties of the nanoplates are described by using a modified power-law function. The Eringen’s nonlocal elasticity is used to capture the size effects. Using the Hamilton’s principle, the governing equations of the porous FG nanoplates using the higher order shear deformation theory are derived. The obtained results demonstrate the significance effect of nonlocal parameter, material composition, porosity factor, porosity distributions, volume fraction exponent and geometrical parameters on static and free vibration analyses of nanoplates.
1. Introduction Materials with porosities as metal foams are one of the most important categories of lightweight materials. The porous volume fraction usually causes a smooth change in mechanical properties. With a fast growth of nanoscale structures, functionally graded materials (FGMs), as specific nanostructures, are called “new revolutionary of materials”. FGMs are often made of two or more materials such as metal and ceramics, which continuously change through the thickness direc tion and have many advantages because of their constituent phases. For example, with ceramic constituents, they can withstand extreme tem perature environments because of their thermal resistance properties, while stronger mechanical performances are found in metal constitu ents. Therefore, FGMs have been used and applied in several engineer ing fields such as civil constructions, biomedical engineering, automotive industry, nuclear engineering, aerospace engineering, etc. Due to their outstanding mechanical, chemical and electrical prop erties, FGM plates/beams with nano or micro length scale have received the attention of researchers and scientists. As we know, the size effect in nanostructures are absent in the classical continuum theory. Therefore, it overpredicts nanostructures behaviors. To overcome these shortcom ings, Eringen (1972) developed the nonlocal elasticity theory based on the continuum mechanics theory by adding a length scale parameter,
which consists of small size effects with good accuracy to nano structures, into the constitutive equations. Based on classical plate the ory (CPT) and finite element method, free vibration analysis of FG nanoplate was reported in Ref. (Natarajan et al., 2012). Buckling behavior of FG nanobeams was conducted in Ref. (Eltaher et al., 2013). Free vibration analysis of FG nanobeams was also examined by Alshorbagy et al. (2011). A refined four-variable plate model for thermal buckling analysis of FG nanoplates under uniform temperature distri butions was investigated by Barati et al. (Barati and Shahverdi, 2016). Besides, a new four-variable plate theory in vibration analysis of FG nanoplates (Belkorissat et al., 2015) was introduced. A new quasi 3D nonlocal plate theory for vibration and buckling of FGM nanoplates was performed in Ref. (Sobhy and Radwan, 2017). Using Navier solutions, free vibration and buckling analyses of FG nanoplate under thermal load (Ansari et al., 2015) were investigated. A nonlocal beam theory for static, buckling and free vibration analyses (Thai, 2012) was reported. Using higher order shear deformation theory, analytical solutions of post-buckling analysis of FG nanoplates considering porosity distribu tions were reported in Ref. (Barati and Zenkour, 2018). Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams using the generalized differential quadrature method (GDQM) (Shafiei et al., 2017) was introduced. Recently, Phung-Van (Phung-Van et al., 2017a, 2017b, 2018) used isogeometric
* Corresponding author. Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam. E-mail address: [email protected] (M. Abdel-Wahab). https://doi.org/10.1016/j.euromechsol.2019.103851 Received 18 August 2018; Received in revised form 22 August 2019; Accepted 6 September 2019 Available online 17 September 2019 0997-7538/© 2019 Elsevier Masson SAS. All rights reserved.
P. Phung-Van et al.
European Journal of Mechanics / A Solids 78 (2019) 103851
Fig. 1. Geometry of FGM nanoplates with two porosity distributions.
Fig. 2. Young’s modulus of porous Al/ZrO2-1.
Table 1 Material properties of FGM. E ⱱ
ρ
Al
ZrO2-2
Ti-Al-4V
SUS304
Al2O3
ZrO2-1
Si3N4
70 � 109 0.3 2707
151 � 109 0.3 3000
320.2 � 109 0.26 3750
201.04 � 109 0.3 8166
380 � 109 0.3 3800
200 � 109 0.3 5700
384.43 � 109 0.3 2370
2
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European Journal of Mechanics / A Solids 78 (2019) 103851
Table 2 Non-dimensional deflections of porous Al/Al2O3 nanoplates with ξ ¼ 0. N
0 2 5
Model
IGA-M (Phung-Van et al., 2017b) IGA-P (Phung-Van et al., 2017b) Present IGA-M (Phung-Van et al., 2017b) IGA-P (Phung-Van et al., 2017b) Present IGA-M (Phung-Van et al., 2017b) IGA-P (Phung-Van et al., 2017b) Present
a/h ¼ 5
a/h ¼ 50
μ
μ
0
1
4
0
1
4
0.0903 0.0903 0.0922 0.2721 0.2284 0.2313 0.3246 0.2856 0.2855
0.1060 0.106 0.1095 0.3192 0.2682 0.2748 0.3803 0.3348 0.3389
0.1530 0.1530 0.1614 0.4605 0.3876 0.4051 0.5476 0.4823 0.4991
0.0750 0.0750 0.0750 0.2222 0.1928 0.1923 0.2564 0.2281 0.2279
0.0886 0.0886 0.0888 0.2625 0.2277 0.2278 0.3028 0.2694 0.2701
0.1294 0.1294 0.1302 0.3834 0.3325 0.3346 0.4422 0.3934 0.3967
Table 3 Porous effects on deflections of Al/Al2O3 nanoplates. N
0
ξ
0.1 0.2 0.3 0.4
1
0.1 0.2 0.3
5
0.1 0.2
10
0.1 0.2
Model
PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II
a/h ¼ 5
a/h ¼ 50
μ
μ
0
1
4
0
1
4
0.0997 0.0941 0.1082 0.0962 0.1177 0.0982 0.1285 0.1004 0.2113 0.1884 0.2559 0.1979 0.3251 0.2085 0.3791 0.3121 0.5888 0.3468 0.4318 0.3527 0.6901 0.3968
0.1185 0.1118 0.1286 0.1142 0.1400 0.1167 0.1529 0.1193 0.2512 0.2238 0.3043 0.2351 0.3869 0.2478 0.4503 0.3705 0.7002 0.4116 0.5127 0.4184 0.8209 0.4705
0.1749 0.1649 0.1900 0.1685 0.2069 0.1722 0.2262 0.1760 0.3708 0.3301 0.4495 0.3469 0.5720 0.3656 0.6639 0.5455 1.0344 0.6060 0.7552 0.6155 1.2108 0.6918
0.0812 0.0765 0.0881 0.0780 0.0958 0.0795 0.1045 0.0811 0.1776 0.1571 0.2170 0.1649 0.2793 0.1737 0.3057 0.2470 0.4875 0.2711 0.3368 0.2706 0.5410 0.2957
0.0962 0.0906 0.1043 0.0924 0.1135 0.0942 0.1238 0.0961 0.2105 0.1862 0.2571 0.1954 0.3311 0.2058 0.3624 0.2928 0.5785 0.3214 0.3992 0.3207 0.6417 0.3505
0.1411 0.1328 0.1531 0.1355 0.1665 0.1382 0.1816 0.1410 0.3090 0.2733 0.3776 0.2869 0.4864 0.3021 0.5325 0.4300 0.8512 0.4723 0.5863 0.4709 0.9435 0.5148
approach for size-dependent analysis of functionally graded carbon nanotube reinforced composites nanoplates (Phung-Van et al., 2017a, 2018) and functionally graded nanoplates (Phung-Van et al., 2017b). In their study, materials without voids and porosities have been examined. In the real materials, nevertheless, there are many porosities and voids, which affect stiffness and reliability of structures. This plays an impor tant role in materials design. Therefore, it motivates us to investigate a new class of materials, i.e. porous materials to give a general and whole view in materials science. By using nonlocal strain theory (NSGT), Li et al. (2015) studied wave propagation in FG nanobeams. Buckling analysis of orthotropic nano plates under thermal environments using exact and differential quad rature methods was reported in Ref. (Farajpour et al., 2016). Analytical solutions of vibration analysis of porous functionally graded nanoplates with attached mass (Shahverdi and Barati, 2017) were recently con ducted. Two length scale parameters were considered in their study. Size-dependent analysis of functionally graded isotropic and sandwich microplates using an improved moving Kriging meshfree (Thai et al., 2018a) and IGA (Thai et al., 2018b) was introduced. Nonlinear and transient isogeometric analysis of FG microplate (Thai et al., 2017) was also reported. It seems that only a few papers were published in porous nanoplates, while understanding its mechanics behavior is highly demanded. All previous studies used analytical solutions to examine nanoplates with
porosities. However, analytical methods are limited to simple problems that can be modelled by partial differential equations. Hence, this paper aims to fill in this gap and makes the first attempt to calculate porous functionally graded nanoplate using numerical methods. The proposed model based on isogeometric analysis and nonlocal continuum theory is developed. Using the nonlocal theory of Eringen, the porousdependency, size-dependency and length scale effect of the nanoplate are performed. Two types of porosity-dependent material properties are incorporated to the modified power-law modeling. The results indicate that behaviors of the nanoplate are significantly influenced by the nonlocal parameter, geometrical parameters and material composition. The obtained results can be considered as benchmark results to analyze porous FG nanoplates. 2. Theoretical formulation 2.1. Porous functionally graded materials We consider a porous nanoplate (length a, width b and thickness h) made of metal and ceramic as indicated in Fig. 1. Two porosity distri butions through thickness of the nanoplate including even porosities (PFGM-I) and uneven porosities (PFGM-II) have been considered as illustrated in Fig. 1. For PFGM-I, porosities are randomly distributed in the cross-section of the nanoplate. Moreover, porosities in PFGM-II are 3
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European Journal of Mechanics / A Solids 78 (2019) 103851
Fig. 3. Young’s modulus of porous Al/Al2O3.
Fig. 5. Deflections of an Al/Al2O3 nanoplate with μ ¼ 1 and n ¼ 1.
Fig. 4. Deflections of an Al/Al2O3 nanoplate with μ ¼ 4 and a/h ¼ 5.
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Table 4 Porous effects on deflections of Al/ZrO2-2 nanoplates with a/h ¼ 10. N
1
ξ
0.1 0.2 0.3 0.4 0.5 0.6
3
0.1 0.2 0.3 0.4 0.5
Model
PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II
Vm þ Vc ¼ 1
;
Vc ¼
μ 0
1
2
3
4
0.3354 0.3036 0.3897 0.3149 0.4623 0.3271 0.5659 0.3402 0.7293 0.3545 1.0394 0.3701 0.3888 0.3473 0.4634 0.3624 0.5709 0.3791 0.7443 0.3976 1.0920 0.4184
0.3986 0.3608 0.4633 0.3743 0.5497 0.3888 0.6729 0.4044 0.8673 0.4214 1.2362 0.4400 0.4621 0.4127 0.5509 0.4308 0.6789 0.4506 0.8853 0.4727 1.2992 0.4975
0.4619 0.4179 0.5368 0.4336 0.6370 0.4505 0.7799 0.4686 1.0054 0.4883 1.4330 0.5099 0.5355 0.4782 0.6385 0.4991 0.7869 0.5222 1.0264 0.5477 1.5064 0.5766
0.5251 0.4751 0.6104 0.4930 0.7243 0.5121 0.8869 0.5328 1.1434 0.5552 1.6298 0.5798 0.6088 0.5436 0.7260 0.5675 0.8949 0.5937 1.1674 0.6228 1.7136 0.6557
0.5883 0.5323 0.6839 0.5523 0.8117 0.5738 0.9939 0.5970 1.2814 0.6221 1.8267 0.6497 0.6822 0.6091 0.8136 0.6359 1.0030 0.6653 1.3084 0.6979 1.9208 0.7348
�n � 1 z þ 2 h
(2)
in which subscript c and m represent ceramic and metal, respectively and n is volume fraction exponent. According to the multi-step sequential infiltration technique, all material properties of PFGM such as density, Poisson’s ratio, Young’s modulus, etc., are defined as: PðzÞ ¼ ðPc
Pm ÞVc þ Pm
PðzÞ ¼ ðPc
Pm ÞVc þ Pm
ξ for PFGM I 2 � � ξ 2jzj 1 for PFGM ðPc þ Pm Þ 2 h ðPc þ Pm Þ
(3) II
From Eq. (3), when ξ is zero, PFGM becomes perfect FGM. Moreover, the expressions of Young’s modulus of PFGM can be formulated as: EðzÞ ¼ ðEc
Em ÞVc þ Em
EðzÞ ¼ ðEc
Em ÞVc þ Em
ξ for PFGM I 2 � � ξ 2jzj 1 for PFGM ðEc þ Em Þ 2 h ðEc þ Em Þ
(4) II
Young’s modulus distributions of Al/ZrO2-1 through the thickness of the nanoplate are also indicated in Fig. 2. We can observe that Young’s modulus without the porosities (ξ ¼ 0) indicated in Fig. 2a is continuous through the top surface (ceramic rich) and the bottom surface (metal rich). Porous effect on Young’s modulus is also performed in Fig. 2b and c. For PFGM-I, forms of curves of Young’s modulus shown in Fig. 2b are the same as those in Fig. 2a with a decrease in Young’s modulus amplitude, leading to decrease in the stiffness of the plate as well. For PFGM-II, the porosities are distributed around the middle zone and tend to zero at the top and the bottom of the cross-section. Therefore, we can see that Young’s modulus shown in Fig. 2c is maximum at the top and the bottom and decreases towards middle zone direction, as clearly shown in Fig. 2d. At the top and the bottom, there is no porosity reached,
distributed around middle zone of the cross-section. Based on the modified rule of mixture, the material properties of porous functionally graded materials (PFGM) are expressed as: � � ξ� ξ� PðzÞ ¼ Pc Vc þ Pm Vm (1) 2 2 where ξ is porosity volume fraction, Vc and Vm are volume fractions of ceramic and metal defined as:
Table 5 The first two non-dimensional natural frequencies of a SSSS SUS304/Si3N4 square nanoplates with a ¼ 10, a/h ¼ 10 and ξ ¼ 0. n
Model
Mode 1
Mode 2
μ
2 10
Ref. (Phung-Van et al., 2017a) Present Ref. (Phung-Van et al., 2017a) Present
μ
0
1
2
4
0
1
2
4
0.0485 0.0466 0.0416 0.0400
0.0443 0.0426 0.0380 0.0365
0.0410 0.0395 0.0352 0.0338
0.0362 0.0349 0.0311 0.0299
0.1154 0.1138 0.0990 0.0975
0.0944 0.0930 0.0810 0.0797
0.0819 0.0806 0.0702 0.0691
0.0669 0.0659 0.0574 0.0564
Fig. 6. Young’s modulus of porous Al/ZrO2-2. 5
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Fig. 9. Deflections of a Ti-Al-4V/Al2O3 nanoplate with n ¼ 1.5, a/h ¼ 20 and μ ¼ 2.
Fig. 7. Deflections of an Al/ZrO2-2 nanoplate (PFGM-I) with n ¼ 1 and a/h ¼ 10.
where αðjx xj; τÞ is the nonlocal kernel function; σij is the local stress tensor at point x, and τ is a length scale parameter. According to Hooke’s law, the local stress is also defined as: 0
(7)
σ ij ¼ Cijkl εkl
where Cijkl is the elasticity tensor. Following Ref. (Eringen, 1972), Eq. (6) can be rewritten as: � 1 μr2 tij ¼ σ ij (8) where μ is a nonlocal parameter and r2 ¼ ∂2 =∂x2 þ ∂2 =∂y2 is the Laplace operator. The governing equations for non-local elastic can be given as: � � σ ij;j þ 1 μr2 fi ¼ 1 μr2 ρu€i (9) To transform a strong form to a weak form, we multiply Eq. (9) with δui and then performing integration over the domain together with a chain of integration, then we obtain the following form: Z Z Z Z � � 1 μr2 ρu€i δui dV ¼ 1 μr2 fi δui dV þ σij nj δui dΓ σ ij δεij dV þ V
Fig. 8. Deflections of an Al/ZrO2-2 nanoplate (PFGM-II) with n ¼ 1 and a/h ¼ 10.
V
Γt
(10)
2.3. Higher order shear deformation theory
i.e. Young’s modulus amplitude of PFGM-II is that of FGM. Moreover, Young’s modulus amplitude of PFGM-II equals to that of PFGM-I at the mid surface.
In equivalent single-layer theories, two theories including classical laminated theory (CLT) and the first-order shear deformation theory (FSDT) are commonly used. However, solutions achieved by CLT and FSDT are not correct because the interlaminar shear deformation in CLT is neglected, while the shear correction factor is required to ensure the stability of the solution of FSDT. To overcome those difficulties, in this study, the higher-order shear deformation theory (HSDT) without the shear correction factor is used. In HSDT, the transverse shear stresses are represented correctly. The displacement field of the porous FGM nano plate based on HSDT can be formulated as follows:
2.2. Nonlocal continuum theory The equations of nonlocal elastic solids are considered herein and formulated as: tij;j þ fi ¼ ρu€i
V
(5)
where i,j represent the symbols z, y, x; fi is the body load and ρ is the density. Based on the nonlocal continuum theory of Eringen (1972), the stress tensor can be expressed as follows: Z 0 0 0 tij ¼ αðjx xj; τÞσij ðx Þdx (6)
uðx; y; zÞ ¼ u0 ðx; yÞ vðx; y; zÞ ¼ v0 ðx; yÞ wðx; y; zÞ ¼ w0 ðx; yÞ
V
6
∂w þ f ðzÞβx ðx; yÞ ∂x ∂w z þ f ðzÞβy ðx; yÞ ∂y z
(11)
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European Journal of Mechanics / A Solids 78 (2019) 103851
Fig. 10. Porous effects on deflections of a FGM nanoplate with n ¼ 1.5, a/h ¼ 20 and μ ¼ 2.
Fig. 11. Young’s modulus of PFGM with n ¼ 1.5, ξ ¼ 0:3 and μ ¼ 2.
Table 6 The six lowest frequencies of a SSSS Al/Al2O3 with n ¼ 3. μ
ξ
Type
0
0 0.1
FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II
0.2 0.3 1
0 0.1 0.2 0.3
2
0 0.1 0.2 0.3
Mode 1
2
3
4
5
6
0.0599 0.0554 0.0592 0.0485 0.0584 0.0346 0.0572 0.0547 0.0506 0.0541 0.0443 0.0533 0.0316 0.0523 0.0507 0.0469 0.0501 0.0411 0.0494 0.0293 0.0484
0.1456 0.1350 0.1437 0.1183 0.1414 0.0845 0.1382 0.1191 0.1104 0.1176 0.0968 0.1156 0.0691 0.1130 0.1032 0.0957 0.1019 0.0839 0.1002 0.0599 0.0980
0.1456 0.1350 0.1438 0.1183 0.1414 0.0845 0.1382 0.1191 0.1104 0.1176 0.0968 0.1157 0.0691 0.1130 0.1032 0.0957 0.1019 0.0839 0.1002 0.0599 0.0980
0.2127 0.2060 0.2099 0.1823 0.2062 0.1311 0.2013 0.1668 0.1548 0.1646 0.1361 0.1618 0.0980 0.1581 0.1388 0.1289 0.1370 0.1134 0.1347 0.0816 0.1316
0.2128 0.2061 0.2099 0.1939 0.2062 0.1619 0.2013 0.1944 0.1809 0.1916 0.1594 0.1879 0.1148 0.1831 0.1589 0.1478 0.1566 0.1302 0.1536 0.0939 0.1496
0.2234 0.2075 0.2205 0.1939 0.2168 0.1628 0.2118 0.1955 0.1820 0.1928 0.1604 0.1892 0.1155 0.1844 0.1598 0.1487 0.1575 0.1310 0.1546 0.0944 0.1507
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Table 7 The six lowest frequencies of a CCCC Al/Al2O3 with n ¼ 3. μ
ξ
Type
0
0 0.1
FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II FGM PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II
0.2 0.3 1
0 0.1 0.2 0.3
2
0 0.1 0.2 0.3
Mode 1
2
3
4
5
6
0.1091 0.1014 0.1078 0.0891 0.1061 0.0635 0.1037 0.0981 0.0911 0.0969 0.0799 0.0953 0.0569 0.0932 0.0898 0.0833 0.0887 0.0731 0.0872 0.0519 0.0853
0.2091 0.1950 0.2062 0.1722 0.2025 0.1243 0.1974 0.1671 0.1556 0.1648 0.1372 0.1617 0.0986 0.1577 0.1430 0.1332 0.1410 0.1173 0.1385 0.0841 0.1350
0.2092 0.1951 0.2064 0.1723 0.2026 0.1244 0.1976 0.1672 0.1557 0.1649 0.1373 0.1619 0.0986 0.1579 0.1431 0.1333 0.1412 0.1173 0.1386 0.0841 0.1351
0.3013 0.2811 0.2971 0.2482 0.2916 0.1791 0.2842 0.2194 0.2043 0.2163 0.1800 0.2122 0.1290 0.2068 0.1808 0.1683 0.1782 0.1481 0.1749 0.1059 0.1704
0.3446 0.3227 0.3393 0.2870 0.3324 0.2108 0.3232 0.2380 0.2226 0.2343 0.1975 0.2295 0.1440 0.2232 0.1927 0.1801 0.1897 0.1597 0.1858 0.1162 0.1807
0.3479 0.3256 0.3426 0.2895 0.3357 0.2123 0.3265 0.2412 0.2254 0.2374 0.1999 0.2326 0.1456 0.2262 0.1958 0.1830 0.1928 0.1621 0.1889 0.1179 0.1837
Table 8 The first natural frequency of a CCCC Al/ZrO2-1 with ξ ¼ 0:3. n
BCs
Type
1
SSSS
PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II PFGM-I PFGM-II
CCCC 5
SSSS CCCC
15
SSSS CCCC
20
SSSS CCCC
μ 0
0.5
1.5
2.5
3
3.5
4
0.0764 0.0837 0.0837 0.1536 0.0755 0.0855 0.1389 0.1545 0.0754 0.0852 0.1388 0.1538 0.0745 0.0845 0.1373 0.1527
0.0728 0.0798 0.0798 0.1452 0.0720 0.0815 0.1314 0.1462 0.0720 0.0812 0.1312 0.1455 0.0711 0.0806 0.1298 0.1445
0.0671 0.0735 0.0735 0.1318 0.0663 0.0750 0.1193 0.1328 0.0662 0.0748 0.1192 0.1323 0.0654 0.0742 0.1179 0.1313
0.0625 0.0685 0.0685 0.1215 0.0618 0.0699 0.1100 0.1225 0.0617 0.0697 0.1099 0.1220 0.0609 0.0691 0.1087 0.1212
0.0605 0.0663 0.0663 0.1172 0.0598 0.0677 0.1060 0.1182 0.0598 0.0675 0.1060 0.1177 0.0590 0.0669 0.1048 0.1169
0.0587 0.0643 0.0643 0.1133 0.0580 0.0657 0.1025 0.1143 0.0580 0.0655 0.1025 0.1138 0.0573 0.0649 0.1014 0.1130
0.0571 0.0625 0.0625 0.1097 0.0564 0.0638 0.0993 0.1107 0.0564 0.0636 0.0993 0.1102 0.0557 0.0631 0.0982 0.1095
where u0, v0 and w0 are the displacements in plane and deflection in the z direction; βx and βy are rotations of the plate and f(z) depends on the
2
�
4 3 plate thickness. In this study, f(z) is chosen as fðzÞ ¼ z z . 3h2 Based on the generalize shear deformation theory, the strains can be expressed as: 3 2 ∂u 6 ∂x 7 7 2 3 6 6 7 εxx 6 ∂v 7 4 5 6 7 ε ¼ εyy ¼ 6 7 6 ∂y 7 εxy 6 7 4 ∂u ∂v 5 þ ∂y ∂x 3 2 2 3 2 3 ∂2 w ∂βx ∂u0 6 2 7 6 7 6 7 6 ∂x 7 ∂x ∂x 6 7 6 7 6 7 6 7 6 ∂v 7 6 ∂2 w 7 ∂ β 6 7 y 6 7 6 7 0 6 7 ¼6 þ f ðzÞ6 (12) 7 þ z6 2 7 7 ∂ y 6 7 6 ∂x ∂ y 7 7 6 6 7 6 7 7 6 4 ∂u0 ∂v0 5 6 7 4 ∂βx ∂βy 5 4 ∂2 w 5 þ þ 2 ∂y ∂x ∂y ∂x ∂x∂y
γ¼
γ xz γ yz
�
3
∂u ∂w � � 6 ∂z þ ∂x 7 β 0 6 7 ¼6 7 ¼ f ðzÞ βx 4 ∂v ∂w 5 y þ ∂z ∂y
(13)
In a shorter form, Eq. (12) can be rewritten as:
ε ¼ εm þ zκ1 þ f ðzÞκ2 ;
0
γ ¼ f ðzÞεs
(14)
where 2
3 3 3 2 2 � � βx;x u0;x w0;xx β εm ¼ 4 v0;y 5 ; κ1 ¼ 4 w0;yy 5; κ2 ¼ 4 βy;y 5 ; εs ¼ x βy βx;y þ βy;x u0;y þ v0;x 2w0;xy
(15)
According to Hooke’s law, the stresses are defined as: �T σ ¼ σ xx σ yy σ xy ¼ Cb ε ¼ Cb ðεm þ zκ1 þ f ðzÞκ2 Þ �T 0 τ ¼ τxz τyz ¼ Cs γ ¼ Cs f ðzÞεs Based on Eq. (8), the stress resultants are formulated as:
8
(16)
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Fig. 12. The lowest six mode shapes of a square porous nanoplate.
8 9 8 8 9 9 8 9 Zt=2 < σ xx = Zt=2 < σ xx = �< Nxx = �< Mxx = 2 Nyy ¼ σ yy dz ; 1 μr Myy ¼ σ yy zdz; 1 μr : ; : : ; ; : ; Nxy σ xy Mxy σ xy t=2 t=2 8 9 8 9 t=2 t=2 � � � � Z Z < σxx = �< Pxx = � Qxz τxz 0 σyy f ðzÞdz ; 1 μr2 1 μr2 Pyy ¼ ¼ f ðzÞdz Qyz τyz : ; : ; Pxy σxy t=2 t=2
2
2
Cb ¼
(17) Substituting Eq. (16) 2 3 2 N A �6 7 6 B 2 6M7 6 1 μr 4 5 ¼ 4 P E Q 0
into Eq. (17), we obtain: 32 3 εm B E 0 6 7 D F 0 7 76 κ1 7 F G 0 54 κ2 5 0 0 As εs
ðA; B; D; E; F; GÞ ¼ Cb Z 0 ¼ Cs ½f ðzÞ�2 dz
6 6 νðzÞ 6 νðzÞ 6 4 0
EðzÞ
νðzÞ
0
1
0
2
1
0
3
7 7 7 7 νðzÞ 5
1
;
Cs ¼
� EðzÞ 1 2ð1 þ νðzÞÞ 0
2
0 1
�
(20)
A weak form of the PFGM nanoplate under transverse load q0 can be expressed as: Z Z � � � € ðδεÞT Dmb ε þ ðδγÞT As γ dΩ þ δuT m 1 μr2 udΩ Ω Ω Z � ¼ δ w μr2 w q0 dΩ (21)
(18)
Ω
where Z
1
where � 1; z; z2 ; f ðzÞ; zf ðzÞ; f 2 ðzÞ dz ;
2
3 A B E 6 7 Dmb ¼ 4 B D F 5; E F H Z ðI1 ; I2 ; I3 ; I4 ; I5 ; I6 Þ ¼
As (19)
in which
2
2 I1 I2 7 6 I 0 5; I ¼ 4 I2 I3 0 I I4 I5 h=2 � ρ 1; z; z2 ; f ðzÞ; zf ðzÞ; f 2 ðzÞ dz I 6 m ¼ 40 0
h=2
and
9
0
0
3
I4
3
7 I5 5 I6
(22)
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3 3 2 2 v0 u0 3 2 3 u1 w0 7 6 6 ∂w 7 7 6 ∂w 7 6 ; u ; u u ¼ 4 u2 5; u1 ¼ 6 ¼ ¼ 7 2 6 7 3 4 0 5 4 ∂x 5 4 ∂y 5 0 u3 βx βy
From Eq. (29), we observe that the basis functions are required at least second and third order derivatives. Hence, IGA can be considered as the most suitable and strong candidate in order to analyze porous nanoplates.
2
(23)
Note that, we can observe that Eq. (21) is in the third order derivative form. Therefore, the order of the basic functions of approximated dis placements is at least a third order. Therefore, isogeometric analysis (IGA) (Cottrell et al., 2009) is a strong and suitable candidate to analyze nanoplates, which due to IGA can easily achieve any desired degree of smoothness through the choice of the interpolation order. Nowadays, IGA has been successfully used to calculate several fields such as com posite plates/shells (Nguyen et al., 2014, 2017; Phung-Van et al., 2015a, 2015b; Tran et al., 2017; Nguyen-Thanh et al., 2015; Thai et al., 2019a, 2019b), large deformation (Nguyen-Thanh et al., 2014, 2017; Vu-Bac et al., 2018), phase-field analysis (Areias et al., 2016), optimization (Lieu and Lee, 2017) and so on.
4. Numerical results
3. Porous functionally graded nanoplate formulation
w¼
Several examples reported in this paper aim to show our new con tributions in functionally graded nanoplates with porosity. Table 1 lists the material properties of functionally graded materials. 4.1. Static analysis We now consider a square nanoplate made of Al/Al2O3 with po rosities. To verify the present results, porous parameter is set zero. Nondimensional deflection is used and defined as follows:
m�n X
(24)
RI ðξ; ηÞdI
I¼1
where dI ¼ ½u0I v0I wI βxI βyI �T is a vector of degrees of freedom of the nanoplate, and RI is the NURBS basis function (Cottrell et al., 2009). Substituting Eq. (24) into Eqs. (14) and (15), the strains are rewritten as:
εm ¼
m�n X
BmI dI ; κ1 ¼
I¼1
m�n X
Bb1 I dI ; κ2 ¼
I¼1
where 2
RI;x 0 6 ¼ 4 0 RI;y RI;y RI;x � 0 0 0 RI BsI ¼ 00 0 0
BmI
3
Bb2 I dI ; εs ¼
I¼1
2 00 7 6 b1 0 0 0 5 ; BI ¼ 4 0 0 00 000 � 0 000
m�n X
m�n X
BsI dI
(25)
I¼1
3 2 3 RI;xx 0 0 0 0 0 RI;x 0 7 7 6 b2 RI;yy 0 0 5 ; BI ¼ 4 0 0 0 0 RI;y 5; 2RI;xy 0 0 0 0 0 RI;y RI;x
RI
(26)
The governing equations for static and free vibration analyses are formulated by: (27)
Kd ¼ F K
�
(28)
ω2 M d ¼ 0
where Z h i � m b1 b2 � � �T B B B Dmb Bm Bb1 Bb2 þ ðBs ÞT As Bs dΩ K¼ ZΩ � ~ μr 2 R ~Tm R ~ dΩ M¼ R ZΩ � R μr2 R q0 dΩ F¼
; Dm ¼
100Em h3 � 12 1 ν2m
(31)
Effect of nonlocal parameters on non-dimensional deflection of the nanoplate is listed in Table 2. Reference solutions were reported in Ref. (Phung-Van et al., 2017b), and the material properties of func tionally graded materials are based on Mori Tanaka (IGA-M) and the mixture rule or power-law (IGA-P). It can be observed that the present results match well with the reference solutions (Phung-Van et al., 2017b) for all cases. Next, porous effects on deflections of Al/Al2O3 nanoplates are listed in Table 3. It can be seen that with an increase of porous parameter, deflections of the nanoplate increase as well. This is because the increase of porous factor leads to a decrease of stiffness of the nanoplate. Besides, effects of porosities on Young’s modulus of Al/Al2O3 are also investi gated in Fig. 3. We can see that with ξ ¼ 0:4 Young’s modulus is nega tive. Therefore, porous factor in this case is only set lower than 0.4. Volume fraction exponent effects on non-dimensional deflection of the PFGM nanoplate with μ ¼ 4 and a/h ¼ 5 are studied and performed in Fig. 4. Again, it can be seen that the deflection of the plate increases with an increase of the volume fraction exponent. Fig. 5 shows the effects of length-thickness ratios (a/h) on deflection of the plate. We observe that deflections decrease with an increase of a/h. Furthermore, porous effects on deflections of Al/ZrO2-2 nanoplates with a/h ¼ 10 are investigated and listed in Table 4. And porous parameter effects on Young’s modulus are examined in Fig. 6 which shows that a porous factor less than 0.6 leads to positive Young’s modulus. Fig. 7 and Fig. 8 show the deflection against nonlocal factor for PFGM-I and PFGM-II nanoplates, respectively. Again, with an increase of porous parameter, deflection increases, as expected. Furthermore, deflections of a Ti-Al-4V/Al2O3 nanoplate with n ¼ 1.5, a/h ¼ 20 and μ ¼ 2 are drawn in Fig. 9. It can be seen that deflection of PFGM-I is larger than that of PFGM-II and FGM. A com parison between deflections of Al/Al2O3, Al/ZrO2-1 and Al/ZrO2-2 is illustrated in Fig. 10. Distributions of Young’s modulus are also shown in Fig. 11. We can see that deflection is the smallest for Al/Al2O3 and the largest for Al/ZrO2-2. This means that the stiffness of Al/Al2O3 plate is the largest and that of Al/ZrO2-2 is the smallest.
Based on NURBS basis functions (Cottrell et al., 2009), the displacement field is defined as follows: uh ðξ; ηÞ ¼
wDm q0 a4
(29)
Ω
with 8 9 2 RI 0 < R1 = ~ R ¼ R2 ; R1 ¼ 4 0 0 : ; 0 0 R3 2 0 0 RI ¼ 40 0 0 0 0 0
4.2. Free vibration 2 3 0 0 0 0 RI 5 RI;x 0 0 ; R2 ¼ 4 0 0 0 0 0 RI 0 3 0 0 0 05 0 0
3 0 0 0 RI;y 0 0 5; R3 0 0 RI
Firstly, a square nanoplate made of SUS304/Si3N4 is investigated. The non-dimensional frequency is used and defined as: rffiffiffiffiffi ρc Ec ; Gc ¼ ω ¼ ωh (32) Gc 2ð1 þ νc Þ The first two frequencies of the nanoplate with a ¼ 10, a/h ¼ 10 and ξ ¼ 0 are listed in Table 5. We can see that the present results match well with those reported in Ref. (Phung-Van et al., 2017a).
(30)
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European Journal of Mechanics / A Solids 78 (2019) 103851
P. Phung-Van et al.
Next, effects of nonlocal and porous parameters on the six lowest frequencies of an Al/Al2O3 nanoplate with simply supported (SSSS) and clamped (CCCC) boundary conditions are presented in Table 6 and Table 7, respectively. It can be observed that an increase of porous parameter leads to a decrease of the stiffness of the plate and the fre quencies increase. Besides, the frequencies decrease with an increase of nonlocal parameter was also reported in Ref. (Phung-Van et al., 2017a). Also, the frequencies of the CCCC nanoplate are higher than those of the SSSS nanoplate. Finally, the effect of the nonlocal parameter on the first natural frequency of the porous nanoplate is shown in Table 8. The frequencies of the nanoplate decrease with an increase of the nonlocal parameter, as expected. The first six mode shapes of the porous nanoplate are also plotted in Fig. 12.
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5. Conclusions We have for the first time presented static and free vibration analyses of nonlocal porous FG nanoplates using higher order shear deformation theory (HSDT). Material properties of the nanoplates are described via a modified power-law function. The Eringen’s nonlocal elasticity is used to analyze the length scale effects. Based on the Hamilton’s principle, the governing equations of the porous FG nanoplates using the gener alized higher order shear deformation theory are obtained. Based on the present formulations numerical results, the following conclusions are drawn: i. Isogeometric analysis of porosity-dependent FG nanoplates using IGA based on HSDT was presented. ii. The integration between the nonlocal theory and IGA is a suit able, strong and effective candidate to study the porous FG nanoplates. iii. Deflection of the plate increases with the increase of the volume fraction exponent. iv. Deflection of PFGM-I is larger than that of PFGM-II with the in crease of porous parameter. v. For free vibration analysis, the increase of porous parameter leads to the decrease of the stiffness of the plate and increase of the frequencies. Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02–2019.09. References Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F.F., 2011. Free vibration characteristics of a functionally graded beam by finite element method. Appl. Math. Model. 35 (1), 412–425. Ansari, R., Ashrafi, M., Pourashraf, T., Sahmani, S., 2015. Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory. Acta Astronaut. 109, 42–51. Areias, P., Rabczuk, T., Msekh, M., 2016. Phase-field analysis of finite-strain plates and shells including element subdivision. Comput. Methods Appl. Mech. Eng. 312, 322–350. Barati, M.R., Shahverdi, H., 2016. A four-variable plate theory for thermal vibration of embedded FG nanoplates under non-uniform temperature distributions with different boundary conditions. Struct. Eng. Mech. 60 (4), 707–727. Barati, M.R., Zenkour, A.M., 2018. Analysis of Postbuckling Behavior of General HigherOrder Functionally Graded Nanoplates with Geometrical Imperfection Considering Porosity Distributions. Mechanics of Advanced Materials and Structures. Belkorissat, I., Houari, M.S.A., Tounsi, A., Bedia, E.A.A., Mahmoud, S.R., 2015. On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model. Steel Compos. Struct. 18 (4), 1063–1081.
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