Probing the shear modulus of two-dimensional multiplanar nanostructures and heterostructures

Abstract

Generalized high-fidelity closed-form formulae are developed to predict the shear modulus of hexagonal graphene-like monolayer nanostructures and nano-heterostructures based on a physically insightful analytical approach. Hexagonal nano-structural forms (top view) are common for nanomaterials with monoplanar (such as graphene, hBN) and multiplanar (such as stanene, MoS2) configurations. However, a single-layer nanomaterial may not possess a particular property adequately, or multiple desired properties simultaneously. Recently a new trend has emerged to develop nano-heterostructures by assembling multiple monolayers of different nanostructures to achieve various tunable desired properties simultaneously. Shear modulus assumes an important role in characterizing the applicability of different two-dimensional nanomaterials and heterostructures in various nanoelectromechanical systems such as determining the resonance frequency of the vibration modes involving torsion, wrinkling and rippling behavior of two-dimensional materials. We have developed mechanics-based closed-form formulae for the shear modulus of monolayer nanostructures and multi-layer nano-heterostructures. New results of shear modulus are presented for different classes of nanostructures (graphene, hBN, stanene and MoS2) and nano-heterostructures (graphene-hBN, graphene-MoS2, graphene-stanene and stanene-MoS2), which are categorized on the basis of the fundamental structural configurations. The numerical values of shear modulus are compared with the results from scientific literature (as available) and separate molecular dynamics simulations, wherein a good agreement is noticed. The proposed analytical expressions will enable the scientific community to efficiently evaluate shear modulus of wide range of nanostructures and nanoheterostructures.

Full text

a∗b c b
2
2
2 2
∗

2 2
S2
2
2
2
2

2
2 2 2 2

2
2 2

2 2 2 2
2
2
α= 0
2
2 2 2
E Eb
EsEtEnb
E=Es+Eb+Et+Enb

2
EbI EbO
E
E=Es+EbI +EbO
=1
2kr(∆l)2+1
2kθ(∆θ)2+1
2kθ(∆α)2
∆l∆θ∆α
krkθ
Es
EbI EbO
krkθ

l A I E
N
Ua=1
2ZL
0
N2
EA dl=1
2
N2l
EA =1
2
EA
l(∆l)2
M∆φ
Ub=1
2ZL
0
M2
EI dl=1
2
EI
l(2∆φ)2
Es
Kr=EA
l
2∆φ∆θ∆α
EbI
EbO kθ=EI
l
krkθ
EA E I
α
M=F l cos α
4

F= 2τl2cos ψcos αsin α
M
δ0=Ml2
6EI
φ=δ0
l
=F l2cos α
24EI
F
δr=1
2φl =F l3cos α
48EI
F
δb=F l3
24EI
us=δb+δr=F l3
48EI (cos α+ 2)
S
S=τl2sin αcos α(1 + sin ψ)τ
F S
S=F(1 + sin ψ)
2 cos ψ
ψ α S F
2
δa=Ssin ψ+F
2cos ψcos αl cos αsin ψ
AE
=F l sin ψ(1 + sin ψ) cos2α
2AE cos ψ
γa=2δa
2lcos ψcos α

γb=us
l(1 + sin ψ) cos α
δaus
γ= 2(γa+γb)
=τl2cos ψcos αsin αsin ψ(1 + sin ψ)
AE cos2ψ+l2(cos α+ 2)
6EI (1 + sin ψ) cos α
EI AE
kθkrKr=EA
lkθ=EI
l
G12 =τ
γ
=krkθcos ψ(1 + sin ψ)
tkθsin ψ(1 + sin ψ)2cos α+krl2
6cos2ψ(cos α+ 2)
ψ= 90◦−θ
2θ
n
τtL =
n
X
i=1
τitiL

G12γt =
n
X
i=1
G12iγiti
G12 γ
G12iγiith
γ=γi,∈[1, n]
G12 =1
t
n
X
i=1
G12iti
=1
t
n
X
i=1
krikθ i cos ψi(1 + sin ψi)
kθi sin ψi(1 + sin ψi)2cos αi+kril2
i
6cos2ψi(cos αi+ 2)
i G12
ith t
˜
G12 =cos ψ(1 + sin ψ)
(sin ψ(1 + sin ψ)2cos α+ 2λcos2ψ(cos α+ 2))

λ(= l2
12
kr
kθ
0.4 2.8
λ4
3l
d2
krkθl d
λ
˜
G12 =G12t
kr
α0α= 0

G12 =krkθcos ψ(1 + sin ψ)
tkθsin ψ(1 + sin ψ)2+krl2
2cos2ψ
θ120◦
ψ= 30◦
G12 =2√3kθkr
t(3kθ+krl2)
n= 1
t
2

krkθ
krkθ
kr= 938 −1−2= 6.52 ×10−7
−1kθ= 126 −1−2= 8.76 ×10−10 −2
α= 0 θ= 120◦ψ= 30◦
krkθ
kr= 4.865 ×10−7−1kθ= 6.952 ×10−10 −2
α= 0 θ= 120◦ψ= 30◦
krkθ
kr= 0.85 ×10−7−1kθ= 1.121 ×10−9−2
α= 17.5◦θ= 109◦ψ= 35.5◦
2krkθ
kr= 1.646 ×10−7−1kθ= 1.677 ×10−9−2
2α= 48.15◦
θ= 82.92◦ψ= 48.54◦
2
¯
G12 =G12 ×t
t
G12 ¯
G12)
t
2

¯
G12) =G12 ×t
t
2
¯
G12 = 0.1254 0.0952±0.0122
¯
G12 = 0.0951
¯
G12 = 0.0325
2¯
G12 = 0.0719
2
2
× ×
2

λ= 0.4λ= 1.2
λ= 2.0λ= 2.8
θ α λ =l2kr
12kθ
˜
G12 =G12t
kr
l t
˙
A
2
G12 θ α
λ
λ0.4
2.8
λ= 1.2507,2.495,0.5061,0.479 2
λ= 0.4,1.2,2.0,2.8

G12 2
2
±
2 2
2
2
2
2

G12
2
2
G12 G12 G12
2

λ= 0.4λ= 1.2
λ= 2.0λ= 2.8
G12 2
2
2

2
2
2

2
2 2

ti2C

X2(x=,)

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