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Effective in-plane elastic moduli of quasi-random spatially irregular hexagonal lattices

June 2017
International Journal of Engineering Science 119

DOI:10.1016/j.ijengsci.2017.06.004

Authors:

Tanmoy Mukhopadhyay

University of Southampton

Sondipon Adhikari

University of Glasgow

An analytical framework is developed for predicting the effective in-plane elastic moduli (longitudinal and transverse Young's modulus, Poisson's ratios and shear modulus) of irregular hexagonal lattices with generalized form of spatially random structural geometry. On the basis of a mechanics based bottom-up multi-step approach, computationally efficient closed-form formulae are derived in this article. As a special case when there is no irregularity, the derived analytical expressions reduce to the respective well known formulae of regular honeycombs available in literature. Previous analytical investigations include the derivation of effective in-plane elastic moduli for hexagonal lattices with spatially random variation of cell angles, which is a special case of the generalized form of irregularity in material and structural attributes considered in this paper. The present study also includes development of a highly generalized finite element code for obtaining equivalent elastic properties of random lattices, which is employed to validate the proposed analytical formulae. The statistical results of elastic moduli obtained using the developed analytical expressions and using direct finite element simulations are noticed to be in good agreement affirming the accuracy and validity of the proposed analytical framework. All the in-plane elastic moduli are found to be significantly influenced by spatially random irregularity resulting in a decrease of the mean values for the two Young's moduli and two Poisson's ratios, while an increase of the mean value for the shear modulus.

Occurrence and application of hexagonal lattices across the length scales

…

Typical representation for (a) a regular hexagonal lattice (b) an irregular hexagonal lattice with spatially random cell angle (c) an irregular hexagonal lattice with generalized form of spatially varying structural randomness

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(a) Typical representation of an irregular honeycomb (b) Representative unit cell element (RUCE) (c) Illustration to dene degree of irregularity (d) Unit cell considered for regular hexagonal lattice by Gibson and Ashby (1999)

…

Idealization of RUCE and proposed bottom-up approach for propagation of the eect of irregularity from an elementary level to the global level

…

+10

RUCE and free-body diagram for the proposed analysis of E 1U

…

Figures - uploaded by Tanmoy Mukhopadhyay

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Content uploaded by Tanmoy Mukhopadhyay

Content may be subject to copyright.

ν12

ν21

l1l2l3α β

γ Es

ZU(ω)ω

Z U

σ1

E1ν12

E2ν21 σ2τ

G12

ZUZI

Z Zeq

Z E1U

i j

i= 1,2,3, ..., m j = 1,2,3, ..., n

ith jth Zij j

jth Zj

rd<min h

2,l

2, l cos θ

r=πr2

d×100

2lcos θ(h+lsin θ)

σ1

E1U

P=σ1Lyb Lyb

M1M2

M1=1

2(P l1sin α−Cl1cos α)

M2=1

2(P l2sin β−Cl2cos β)

C=Pl1sin α−l2sin β

l1cos α−l2cos β

δh

P C

δh

AO =P l3

1sin α

12EsI−Cl3

1cos α

12EsIsin α

P C h

I I =bt3/12 t

E1U

δh

BO =P l3

2sin β

12EsI−Cl3

1cos β

12EsIsin β

δO=l2sin βl1cos α−l1sin αl2cos β

l1cos α−l2cos β

δh

1=δh

δO

l1sin α+δh

δO

l2sin β

=σ1Lyl2

1l2

2(l1+l2) (cos αsin β−sin αcos β)2

Est3(l1cos α−l2cos β)2

h

1=σ1Lyl2

1l2

2(l1+l2) (cos αsin β−sin αcos β)2

Est3(l1cos α−l2cos β)3

E1U=Est3(l1cos α−l2cos β)3

Lyl2

1l2

2(l1+l2) (cos αsin β−sin αcos β)2

E1U

1ULjjth

P Bij

AN I E1U

=P Bij

AIEI

ANI =Lyb AI=Ljb

1U=E1U

E1eq σ1

jth

σ1∆1j

∆1ij

jth

∆1j=

i=1

∆1ij

1jBj=

i=1

1ij Bij

1jBjjth

Bij = (l1ij cos αij −l2ij cos βij )Bj=

i=1 Bij

σ1Bj

E1j

i=1

σ1Bij

1Uij

jth ˆ

E1j

E1j=Bj

i=1

Bij

1Uij

i j

E1eq

E1j

σ1Lb =

j=1

σ1jLjb

Ljjth L=

j=1

E1eqL=

j=1

E1jLj

E1eq

E1eq =1

j=1

BjLj

i=1

Bij

1Uij

E1eq =Est3

j=1

i=1

(l1ij cos αij −l2ij cos βij )

i=1

1ij l2

2ij (l1ij +l2ij ) (cos αij sin βij −sin αij cos βij )2

(l1ij cos αij −l2ij cos βij )2

σ2

E2U

δv

bCO =W l3

3cos2γ

12EsI

W=σ2b(l1cos α−l2cos β)v

E2U

M0=−W l3cos γ

2M0

OA =l2

l1+l2

M0M0

OB =l1

l1+l2

l M

δ=Ml2

6EI φ=−M0

OAl1

6EsI

OA M0

δv

rCO =l1l2l2

3cos2γ

12EsI(l1+l2)W

1=W1l1cos α

2+l2

l1+l2

M0

M00

1=W1l1cos α

2−l2

l1+l2

M0

2=−W2l2cos β

2−l1

l1+l2

M0

M00

2=−W2l2cos β

2+l1

l1+l2

M0

W1=−l2cos β

l1cos α−l2cos βW

W2=l1cos α

l1cos α−l2cos βW

δv

AO =W1l3

1cos2α

12EsI

δv

BO =W2l3

2cos2β

12EsI

δv

O=δAO(−l2cos β) + δB O (l1cos α)

l1cos α−l2cos β

δv

AO δv

δv

O=l2

1l2

2cos2αcos2β(l1+l2)

12EsI(l1cos α−l2cos β)2W

δv

2=δv

bCO +δv

rCO +δv

12EsIl2

3cos2γl3+l1l2

l1+l2+l2

1l2

2(l1+l2) cos2αcos2β

(l1cos α−l2cos β)2

v

2=σ2(l1cos α−l2cos β)

Est3Lyl2

3cos2γl3+l1l2

l1+l2+l2

1l2

2(l1+l2) cos2αcos2β

(l1cos α−l2cos β)2

E2U=Est3Ly

(l1cos α−l2cos β)l2

3cos2γl3+l1l2

l1+l2+l2

1l2

2(l1+l2) cos2αcos2β

(l1cos α−l2cos β)2−1

E2ULy

2ULj

jth EI

W Ly

AE2U

=W Lj

AEI

A=Bij b

2U=E2U

E2eq EI

σ2

jth σ2

σ2Bb = m

i=1

σ2ij Bij !b

2ij

jth

E2j2jBj=

i=1

2U ij 2ij Bij

2ij =2ji= 1,2...m jth ˆ

E2j

jth

E2j=

i=1

2U ij Bij

∆2j

g

2L=

j=1

2jLj

g

22jjth

Ljjth

σ2L

E2eq

j=1

σ2Lj

E2j

E2eq =L

j=1

LjBj

i=1

2U ij Bij

E2eq =LEst3

j=1

i=1

(l1ij cos αij −l2ij cos βij )

i=1 l2

3ij cos2γij l3ij +l1ij l2ij

l1ij +l2ij +l2

1ij l2

2ij (l1ij +l2ij ) cos2αij cos2βij

(l1ij cos αij −l2ij cos βij )2−1

G12Uτ

δs

OA =δs

F1=cot α

cot α−cot βF

F2=−cot β

cot α−cot βF

Vs=1

cot α−cot βF

δs

OA =F1sin αl3

12EsI−Vscos αl3

12EsIsin α

s F1

G12U

Vsδs

OA =δs

OB = 0

δs

bCO =F l3

3sin2γ

12EsI

F=τ b (l1cos α−l2cos β)Ms=F l3sin γ

2Ms

MsMs

1=l2

l1+l2

MsMs

2=l1

l1+l2

M δ =Ml2

6EI

φ=l1l2Ms

6EsI(l1+l2)

δs

rCO =φl3sin γ=l1l2l2

3sin2γF

12EsI(l1+l2)

δs

CO =δs

bCO +δs

rCO =F l2

3sin2γ

12EsIl3+l1l2

l1+l2

γs=τ(l1cos α−l2cos β)l2

3sin2γ

Est3Lyl3+l1l2

l1+l2

G12U=τ

γs

=Est3Ly

3sin2γ(l1cos α−l2cos β)l3+l1l2

l1+l2

G12ULy

12U

Ljjth

12U

G12U

Ly=τ

12U

12U=G12U

G12eq τ jth

τBj=

i=1

τij Bij

G12jγjBj=

i=1

12U ij γij Bij

G12jjth γjγij

jth jth GI

12Uij

i j

γj=γij i= 1,2, ..., m jth

G12j=

i=1

12U ij Bij

γgL=

j=1

γjLj

γgG12

G12eq

j=1

τj

G12j

τ=τj

G12eq =L

j=1

LjBj

i=1

12U ij Bij

12Uij

G12eq =LEst3

j=1

i=1

(l1ij cos αij −l2ij cos βij )

i=1 l2

3ij sin2γij l3ij +l1ij l2ij

l1ij +l2ij −1

ν12

ν12U

ν12U=−0

0

10

0

1(= h

δ0

AO =−P l3

1sin α

12EsI−Cl3

1cos α

12EsIcos α

P C

δ0

BO =P l3

2sin β

12EsI−Cl3

1cos β

12EsIcos β

δ0

O=δ0

AO(−l2cos β) + δ0

BO (l1cos α)

l1cos α−l2cos β

=σ1l2

1l2

2Ly(l1+l2) cos αcos β(cos αsin β−sin αcos β)

Est3(l1cos α−l2cos β)2

0

2=σ1l2

1l2

2(l1+l2) cos αcos β(cos αsin β−sin αcos β)

Est3(l1cos α−l2cos β)2

ν12U=−cos αcos β(l1cos α−l2cos β)

(cos αsin β−sin αcos β)Ly

ν12ULy

νI

12U

Ljjth

νI

12U0

1Bij =0I

1Bij

0

2Ly=0I

2Lj(.)I

νI

12U=ν12U

ν12

ν12eq νI

12U

ν12eq σ1

σ1jth

ν12

−2j

ˆν12j

Bj=−

i=1

2ij Bij

νI

U12ij

2j2ij jth jth

νI

U12ij

i j ˆν12jjth

2j=2ij i= 1,2, ..., m jth

ˆν12j=Bj

i=1

Bij

νI

12Uij

σ1

g12

2L=

j=1

2jLj

g12

2ν12

ν12eqg12

1L=

j=1

ν12j1jLj

g12

11jjth

g12

1=1jj= 1,2, ..., n

ν12eq =1

j=1

BLj

i=1

Bij

νI

12Uij

νI

12Uij

ν12eq =−1

j=1

i=1

(l1ij cos αij −l2ij cos βij )

i=1

(cos αij sin βij −sin αij cos βij )

cos αij cos βij

ν21

ν21U

ν21U=−00

00

100

00

2(= v

00

δ00

AO =−W1l3

1cos αsin α

12EsI

δ00

BO =W2l3

2cos βsin β

12EsI

W1W2

δ00

1=δ00

δO

l1sin α+δ00

δO

l2sin β

=W l2

1l2

2(l1+l2) cos αcos β(cos αsin β−sin αcos β)

12EsI(l1cos α−l2cos β)2

δO

00

1=σ2l2

1l2

2(l1+l2) cos αcos β(cos αsin β−sin αcos β)

Est3(l1cos α−l2cos β)2

ν21U=−Lyl2

1l2

2(l1+l2) cos αcos β(cos αsin β−sin αcos β)

(l1cos α−l2cos β)3l2

3cos2γl3+l1l2

l1+l2+l2

1l2

2(l1+l2) cos2αcos2β

(l1cos α−l2cos β)2

ν21ULy

νI

21U

Ljjth

νI

21U00

1Bij =00I

1Bij

00

2Ly=00I

2Lj(.)I

νI

21U=ν21U

ν21

ν21eq νI

21U

σ2ν21eq

E2eq σ2jth

jth

1jBj=

i=1

1ij Bij

ν21

ˆν21j2jBj=

i=1

νI

21U ij 2ij Bij

ˆν21jjth 2j2ij

jth jth

νI

21Uij

i j 2j=2ij i= 1,2, ..., m jth

ˆν21j=

i=1

νI

21U ij Bij

σ2

g21

2L=

j=1

2jLj

g21

2ν21

g21

ν21eq

j=1

1j

ˆν21j

g21

1=1jj= 1,2, ..., n

ν21eq =L

j=1

i=1

νI

21U ij Bij

νI

21Uij

ν21eq =−L

j=1

i=1

(l1ij cos αij −l2ij cos βij )

i=1

1ij l2

2ij (l1ij +l2ij ) cos αij cos βij (cos αij sin βij −sin αij cos βij )

(l1ij cos αij −l2ij cos βij )2l2

3ij cos2γij l3ij +l1ij l2ij

l1ij +l2ij +l2

1ij l2

2ij (l1ij +l2ij ) cos2αij cos2βij

(l1ij cos αij −l2ij cos βij )2

βij >90◦

L=n(h+lsin θ)l1ij =l2ij =l3ij =l αij =θ βij = 180◦−θ γij = 90◦

i j

θ= 30◦

E∗

=E∗

= 2.3t

l3

E∗

1E∗

E1E2G12 ν12 ν21

E1GA =

Est

l3cos θ

l+sin θ) sin2θ

E2GA =Est

l3(h

l+sin θ)

cos3θ

G12GA =

Est

l3(h

l+sin θ)

l)2(1+2 h

l) cos θ

ν12GA =cos2θ

l+sin θ) sin θ

ν21GA =(h

l+sin θ) sin θ

cos2θ

E1eq =Est3

j=1

i=1

(l1ij cos αij −l2ij cos βij )

i=1

1ij l2

2ij (l1ij +l2ij ) (cos αij sin βij −sin αij cos βij )2

(l1ij cos αij −l2ij cos βij )2

E2eq =LEst3

j=1

i=1

(l1ij cos αij −l2ij cos βij)

i=1 l2

3ij cos2γij l3ij +l1ijl2ij

l1ij+l2ij +l2

1ijl2

2ij(l1ij +l2ij ) cos2αij cos2βij

(l1ij cos αij−l2ij cos βij )2−1

G12eq =LEst3

j=1

i=1

(l1ij cos αij −l2ij cos βij)

i=1 l2

3ij sin2γij l3ij +l1ijl2ij

l1ij+l2ij −1

ν12eq =−1

j=1

i=1

(l1ij cos αij −l2ij cos βij )

i=1

(cos αij sin βij −sin αij cos βij )

cos αij cos βij

ν21eq =−L

j=1

i=1

(l1ij cos αij −l2ij cos βij)

i=1

1ijl2

2ij (l1ij +l2ij) cos αij cos βij (cos αij sin βij −sin αij cos βij )

(l1ij cos αij −l2ij cos βij)2l2

3ij cos2γij l3ij +l1ijl2ij

l1ij+l2ij +l2

1ijl2

2ij(l1ij +l2ij ) cos2αij cos2βij

(l1ij cos αij−l2ij cos βij )2

θ= 30◦

G∗

= 0.57 t

l3

G∗

E∗

2ν∗

12 =E∗

1ν∗

21 ν∗

12 ν∗

ν∗

12 =ν∗

21 = 1 G=E/2(1 + ν)

E G ν

ith jth

E1Uij =Esij t3(l1ij cos αij −l2ij cos βij )3

Lyl2

1ij l2

2ij (l1ij +l2ij ) (cos αij sin βij −sin αij cos βij )2

Esij ith jth

E1eq =t3

j=1

i=1

(l1ij cos αij −l2ij cos βij )

i=1

1ij l2

2ij (l1ij +l2ij ) (cos αij sin βij −sin αij cos βij )2

Esij ((l1ij cos αij −l2ij cos βij )2)

E2eq =Lt3

j=1

i=1

(l1ij cos αij −l2ij cos βij )

i=1

Esij l2

3ij cos2γij l3ij +l1ij l2ij

l1ij +l2ij +l2

1ij l2

2ij (l1ij +l2ij ) cos2αij cos2βij

(l1ij cos αij −l2ij cos βij )2−1

G12eq =Lt3

j=1

i=1

(l1ij cos αij −l2ij cos βij )

i=1

Esij l2

3ij sin2γij l3ij +l1ij l2ij

l1ij +l2ij −1

L=n(h+lsin θ)l1ij =l2ij =l3ij =l αij =θ βij = 180◦−θ γij = 90◦i j

E1eq =κ1t

l3cos θ

l+ sin θ) sin2θ

E2eq =κ2t

l3(h

l+ sin θ)

cos3θ

G12eq =κ2t

l3h

l+ sin θ

h

l2(1 + 2h

l) cos θ

κ1κ2

κ1=m

j=1

i=1

Esij

κ2=n

j=1

i=1

Esij

ν12eq =ν12GA ν21eq =ν21GA

Esij Esij =Es

i= 1,2,3, ..., m j = 1,2,3, ..., n κ1κ2Es

30◦h/l = 1

l1l2l3α β γ t

l1l2l3α β γ

h=l θ = 30◦rd= 0.257l r = 8

h l

rd= 0.257l

h/l t/l ∼10−2

h/l

θ◦ ◦ ◦

Esρ3

E1E2ν12 ν21 G12

E1=E1eq

Esρ3¯

E2=E2eq

Esρ3¯ν12 =ν12eq ¯ν21 =ν21eq ¯

G12 =G12eq

Esρ3(¯)

θ= 30◦;h/l = 1

1.3 1.35 1.4 1.45 1.5 1.55 1.6

Density

Analytical

F EM

θ= 30◦h/l = 1 r= 8

h/l = 1 h/l = 1.5

h/l = 2

E1θ= 30◦

h/l = 1 h/l = 1.5

h/l = 2

E1θ= 45◦

h/l = 1 h/l = 1.5

h/l = 2

E1θ= 60◦

E2ν12 ν21

G12

E1G12

θ E1E2

ν12 ν21 G12

h/l

% % E1E2%

h/l = 1 h/l = 1.5

h/l = 2

E2θ= 30◦

h/l = 1 h/l = 1.5

h/l = 2

E2θ= 45◦

h/l = 1 h/l = 1.5

h/l = 2

E2θ= 60◦

G12 % % ν12 ν21

∆m

∆mr

h/l = 1 h/l = 1.5

h/l = 2

G12 θ= 30◦

h/l = 1 h/l = 1.5

h/l = 2

G12 θ= 45◦

h/l = 1 h/l = 1.5

h/l = 2

G12 θ= 60◦

h/l = 1 h/l = 1.5

h/l = 2

ν12 θ= 30◦

h/l = 1 h/l = 1.5

h/l = 2

ν12 θ= 45◦

h/l = 1 h/l = 1.5

h/l = 2

ν12 θ= 60◦

E1E2

G12

h/l = 1 h/l = 1.5

h/l = 2

ν21 θ= 30◦

h/l = 1 h/l = 1.5

h/l = 2

ν21 θ= 45◦

h/l = 1 h/l = 1.5

h/l = 2

ν21 θ= 60◦

E2ν21 G12

E1ν12

E1E2

ν12 ν21 G12

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Analytical investigations to characterize the effective mechanical properties of lattice materials allow an in-depth exploration of the parameter space efficiently following an insightful, yet elegant framework. 2D lattice materials, which have been extensively dealt with in the literature following analytical as well as numerical and experimental approaches, have limitations concerning multi-directional stiffness and Poisson's ratio tunability. The primary objective of this paper is to develop mechanics-based formulations for a more complex analysis of 3D lattices, leading to a physically insightful analytical approach capable of accounting the beam-level mechanics of pre-existing intrinsic stresses along with their interaction with 3D unit cell architecture. We have investigated the in-plane and out-of-plane effective elastic properties to portray the physics behind the deformation of 3D lattices under externally applied far-field normal and shear stresses. The considered effect of beam-level intrinsic stresses therein can be regarded as a consequence of inevitable temperature variation, pre-stress during fabrication, inelastic and non-uniform deformation, manufacturing irregularities etc. Such effects can notably impact the effective elastic properties of lattice materials, quantifying which for 3D honeycombs is the central focus of this work. Further, from the material innovation viewpoint, the intrinsic stresses can be deliberately introduced to expand the microstructural design space for effective elastic property modulation of 3D lattices. This will lead to programming of effective properties as a function of intrinsic stresses without altering the microstructural geometry and lattice density. We have proposed a generic spectral framework of analyzing 3D lattices analytically, wherein the beam-level stiffness matrix including the effect of bending, axial, shear and twisting deformations along with intrinsic stresses can be coupled with the unit cell mechanics for obtaining the effective elastic properties.

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Oct 2024

After tremendous progress in computationally conceptualizing and manufacturing lattice metamaterials with complex cell geometries over the last decade, a strong rationale has evolved lately to achieve active and on-demand property modulation in real-time with greater sensitivity. Traditionally materials show an uncoupled response between normal and shear modes of deformation. Here we propose to achieve heterogeneous mode coupling among the normal and shear modes, but in conventional symmetric lattice geometries through an intuitive mounting of electro-active elements. The proposed bi-level multi-physically architected metamaterials lead to an unprecedented programmable voltage-dependent normal-shear constitutive mode coupling and active multi-modal stiffness modulation capability for critically exploitable periodic or aperiodic, on-demand and temporally tunable mechanical responses. Further, active partial cloaking concerning the effect of far-field complex stresses can be achieved, leading to the prospect of averting a range of failure and serviceability conditions. The manufacturing flexibility in terms of symmetric lattice geometry, along with actively tunable heterogeneous mode coupling in the new class of metamaterials would lead to real-time control of mechanical responses for temporal programming in a wide range of advanced mechanical applications, including morphing and transformable geometries, locomotion in soft robotics, embedded actuators, enhanced multi-modal energy harvesting, vibration and wave propagation control.

Analysis of effective elastic parameters of natural bamboo honeycomb cell structure

Article

Full-text available

Feb 2024
WOOD SCI TECHNOL

Bamboo is an environmentally friendly building structural material. This work investigated the cavity structural characteristics and physical and mechanical properties of honeycomb sandwiches and natural bamboo in the longitudinal direction. The effective elastic parameters of periodically arranged hexagonal bamboo honeycomb cells under in-plane and out-of-plane loads were modeled using analytical and numerical approaches. Then, the effective elastic parameter model of bamboo honeycomb cells was validated by experiments and finite element analysis. The average errors between the calculated and experimental equivalent modulus of elasticity, Poisson’s ratio, and shear modulus in the three principal axis directions were 7.43, 4.37, and 8.68%, respectively. The average relativities between the model values of the elastic parameters of the bamboo honeycomb cell and the simulation results in the three directions were 5.46, 5.40, and 6.12%, respectively. The experimental and finite element analysis showed that the constructed effective elastic parameter model of the bamboo honeycomb cell better reflected the state of the bamboo core when subjected to force. This study provides insights for further research on the mechanical properties of bamboo materials and their application in bamboo-based lightweight and high-strength sandwich structures.

On exploiting the architecture of annual ring growth for developing a new class of bio-inspired composites

Article

Full-text available

Nov 2024
MECH BASED DES STRUC

Inspired by several biological structures available in nature, bio-inspired composite structures are evidenced to exhibit a noteworthy enhancement in various mechanical and multi-physical performances as compared to conventional structures. This article proposes to exploit the architecture of annual ring growth of the stems of trees for developing a new class of bio-inspired composites with enhanced static and dynamic performances, including deflections, stresses, strain energy, and vibration. Concentric circular annual-ring geometries are considered where each layer of concentric circular fibers is analogous to the growth per annum of trees. The annual rings are modeled in a finite element-based computational framework by idealizing each layer as a composite of graphite fibers and epoxy matrix under different boundary conditions. The ratio of deflection to weight and frequency to weight of bio-inspired and traditional composites are compared by considering different parameters such as the number of annual rings, layers, and supporting stiffeners. The numerical results reveal that the proposed bio-inspired composites can enhance and modulate the static and dynamic properties to a significant extent, opening new design pathways for developing high-performance fiber network composites.

Multiscale computational modeling techniques in study and design of 2D materials: recent advances, challenges, and opportunities

Article

Full-text available

Sep 2024

This article provides an overview of recent advances, challenges, and opportunities in multiscale computational modeling techniques for study and design of two-dimensional (2D) materials. We discuss the role of computational modeling in understanding the structures and properties of 2D materials, followed by a review of various length-scale models aiding in their synthesis. We present an integration of multiscale computational techniques for study and design of 2D materials, including density functional theory, molecular dynamics, phase-field modeling, continuum-based molecular mechanics, and machine learning. The study focuses on recent advancements, challenges, and future prospects in modeling techniques tailored for emerging 2D materials. Key challenges include accurately capturing intricate behaviors across various scales and environments. Conversely, opportunities lie in enhancing predictive capabilities to accelerate materials discovery for applications spanning from electronics, photonics, energy storage, catalysis, and nanomechanical devices. Through this comprehensive review, our aim is to provide a roadmap for future research in multiscale computational modeling and simulation of 2D materials.

Hexagonal quasiperiodic tilings as decorations of periodic lattices

Preprint

Full-text available

Apr 2024

Sam Coates

Symmetry sharing facilitates coherent interfaces which can transition from periodic to quasiperiodic structures. Motivated by the design and construction of such systems, we present hexagonal quasiperiodic tilings with a single edge-length which can be considered as decorations of a periodic lattice. We introduce these tilings by modifying an existing family of golden-mean trigonal and hexagonal tilings, and discuss their properties in terms of this wider family. Then, we show how the vertices of these new systems can be considered as decorations or sublattice sets of a periodic triangular lattice. We conclude by simulating a simple Ising model on one of these decorations, and compare this system to a triangular lattice with random defects.

Active mechanical cloaking for unsupervised damage resilience in programmable elastic metamaterials

Article

Full-text available

Jul 2024
PHILOS T R SOC A

Owing to the architected void-filled low-density configurations, metamaterials are prone to defects during the complex manufacturing process, or damages under operational conditions. Recently mechanical cloaking has been proposed to shield the effect of such disorders in terms of homogenized mechanical responses. The major drawback in these studies are that the damage location should be known a priori, and the cloak is designed around that damaged zone before manufacturing. Such postulation does not allow unsupervised damage resilience during the manufacturing and service life of metamaterials by active reconfiguration of the stress field depending on the random and unpredictable evolution of damage. Here, we propose a radically different approach by introducing piezoelectric lattices where the effect of random appearance of any single or multiple disorders and damages with complex shapes, sizes and distributions can be shielded through active multi-physically controlled cloaks by voltage-dependent modulation of the stress fields within the cloaking region. Notably, this can be achieved without breaking periodicity and any additional material in the cloaking region unlike earlier studies concerning mechanical cloaks. The proposed active class of elastic metamaterials will bring a step-change in the on-demand mechanical performance of critically important structural components and unsupervised damage resilience for enhanced durability and sustainability. This article is part of the theme issue ‘Current developments in elastic and acoustic metamaterials science (Part 1)’.

Nonlinear elasticity tailoring and failure mode manipulation of functionally graded honeycombs under large deformation

Article

Oct 2024
INT J NONLIN MECH

Analytical homogenization for equivalent in-plane elastic moduli of prestressed lattices based on the micropolar elasticity model

Article

Jul 2024
COMPOS STRUCT

Effect of damage evolution on the auxetic behavior of 2D and 3D re-entrant type geometries

Article

Mar 2024
MECH MATER

Effective elastic properties of two dimensional multiplanar hexagonal nanostructures

Article

Full-text available

Jan 2017

A generalized analytical approach is presented to derive closed-form formulae for the elastic moduli of hexagonal multiplanar nano-structures. Hexagonal nano-structural forms are common for various materials. Four different classes of materials (single layer) from a structural point of view are proposed to demonstrate the validity and prospective application of the developed formulae. For example, graphene, an allotrope of carbon, consists of only carbon atoms to form a honeycomb like hexagonal lattice in a single plane, while hexagonal boron nitride (hBN) consists of boron and nitrogen atoms to form the hexagonal lattice in a single plane. Unlike graphene and hBN, there are plenty of other materials with hexagonal nano-structures that have the atoms placed in multiple planes such as stanene (consists of only Sn atoms) and molybdenum disulfide (consists of two different atoms: Mo and S). The physics based high-fidelity analytical model developed in this article are capable of obtaining the elastic properties in a computationally efficient manner for wide range of such materials with hexagonal nano-structures that are broadly classified in four classes from structural viewpoint. Results are provided for materials belonging to all the four classes, wherein a good agreement between the elastic moduli obtained using the proposed formulae and available scientific literature is observed.

Stochastic mechanics of metamaterials

Article

Full-text available

Nov 2016
COMPOS STRUCT

The effect of stochasticity in mechanical behaviour of metamaterials is quantified in a probabilistic framework. The stochasticity has been accounted in the form of random material distribution and structural irregularity, which are often encountered due to manufacturing and operational uncertainties. An analytical framework has been developed for analysing the effective stochastic in-plane elastic properties of irregular hexagonal structural forms with spatially random variations of cell angles and intrinsic material properties. Probabilistic distributions of the in-plane elastic moduli have been presented considering both randomly homogeneous and randomly inhomogeneous stochasticity in the system, followed by an insightful comparative discussion. The ergodic behaviour in spatially irregular lattices is investigated as a part of this study. It is found that the effect of random micro-structural variability in structural and material distribution has considerable influence on mechanical behaviour of metamaterials.

Probabilistic Analysis and Design of HCP Nanowires: An Efficient Surrogate Based Molecular Dynamics Simulation Approach

Article

Full-text available

Nov 2016
J MATER SCI TECHNOL

We investigate the dependency of strain rate, temperature and size on yield strength of hexagonal close packed (HCP) nanowires based on large-scale molecular dynamics (MD) simulation. A variance-based analysis has been proposed to quantify relative sensitivity of the three controlling factors on the yield strength of the material. One of the major drawbacks of conventional MD simulation based studies is that the simulations are computationally very intensive and economically expensive. Large scale molecular dynamics simulation needs supercomputing access and the larger the number of atoms, the longer it takes time and computational resources. For this reason it becomes practically impossible to perform a robust and comprehensive analysis that requires multiple simulations such as sensitivity analysis, uncertainty quantification and optimization. We propose a novel surrogate based molecular dynamics (SBMD) simulation approach that enables us to carry out thousands of virtual simulations for different combinations of the controlling factors in a computationally efficient way by performing only few MD simulations. Following the SBMD simulation approach an efficient optimum design scheme has been developed to predict optimized size of the nanowire to maximize the yield strength. Subsequently the effect of inevitable uncertainty associated with the controlling factors has been quantified using Monte Carlo simulation. Though we have confined our analyses in this article for Magnesium nanowires only, the proposed approach can be extended to other materials for computationally intensive nano-scale investigation involving multiple factors of influence.

A multivariate adaptive regression splines based damage identification methodology for web core composite bridges including the effect of noise

Article

Full-text available

Jan 2016

Tanmoy Mukhopadhyay

A novel computationally efficient damage identification methodology for web core fiber-reinforced polymer composite bridges has been developed in this article based on multivariate adaptive regression splines in conjunction with a multi-objective goal-attainment optimization algorithm. The proposed damage identification methodology has been validated for several single and multiple damage cases. The performance of the efficient multivariate adaptive regression splines-based approach for the inverse system identification process is found to be quite satisfactory. An iterative scheme in conjunction with the multi-objective optimization algorithm coupled with multivariate adaptive regression splines is proposed to increase damage identification accuracy. The effect of noise on the proposed damage identification algorithm has also been addressed subsequently using a probabilistic framework. The multivariate adaptive regression splines-based damage identification algorithm is general in nature; therefore, in future it can be implemented to other structures.

Auxetic Materials: Functional Materials and Structures from Lateral Thinking!

Article

May 2000
ADV MATER

Materials that become thicker when stretched and thinner when compressed are the subject of this review. The theory behind the counterintuitive behavior of these so-called auxetic materials is discussed, and examples and applications are examined. For example, blood vessels made from an auxetic material will tend to increase in wall thickness (rather than decrease) in response to a pulse of blood, thus preventing rupture of the vessel (see Figure).

Mathematical Statistics and Data Analysis

Article

Aug 1989
TECHNOMETRICS

Metamodel based high-fidelity stochastic analysis of composite laminates: A concise review with critical comparative assessment

Article

Jan 2017
COMPOS STRUCT

This paper presents a concise state-of-the-art review along with an exhaustive comparative investigation on surrogate models for critical comparative assessment of uncertainty in natural frequencies of composite plates on the basis of computational efficiency and accuracy. Both individual and combined variations of input parameters have been considered to account for the effect of low and high dimensional input parameter spaces in the surrogate based uncertainty quantification algorithms including the rate of convergence. Probabilistic characterization of the first three stochastic natural frequencies is carried out by using a finite element model that includes the effects of transverse shear deformation based on Mindlin’s theory in conjunction with a layer-wise random variable approach. The results obtained by different metamodels have been compared with the results of traditional Monte Carlo simulation (MCS) method for high fidelity uncertainty quantification. The crucial issue regarding influence of sampling techniques on the performance of metamodel based uncertainty quantification has been addressed as an integral part of this article.

Stochastic natural frequency analysis of damaged thin-walled laminated composite beams with uncertainty in micromechanical properties

Article

Oct 2016
COMPOS STRUCT

This paper presents a stochastic approach to study the natural frequencies of thin-walled laminated composite beams with spatially varying matrix cracking damage in a multi-scale framework. A novel concept of stochastic representative volume element (SRVE) is introduced for this purpose. An efficient radial basis function (RBF) based uncertainty quantification algorithm is developed to quantify the probabilistic variability in free vibration responses of the structure due to spatially random stochasticity in the micro-mechanical and geometric properties. The convergence of the proposed algorithm for stochastic natural frequency analysis of damaged thin-walled composite beam is verified and validated with original finite element method (FEM) along with traditional Monte Carlo simulation (MCS). Sensitivity analysis is carried out to ascertain the relative influence of different stochastic input parameters on the natural frequencies. Subsequently the influence of noise is investigated on radial basis function based uncertainty quantification algorithm to account for the inevitable variability for practical field applications. The study reveals that stochasticity/ system irregularity in structural and material attributes affects the system performance significantly. To ensure robustness, safety and sustainability of the structure, it is very crucial to consider such forms of uncertainties during the analysis.

Effect of cutout on stochastic natural frequency of composite curved panels

Article

Aug 2016
COMPOS PART B-ENG

The present computational study investigates on stochastic natural frequency analyses of laminated composite curved panels with cutout based on support vector regression (SVR) model. The SVR based uncertainty quantification (UQ) algorithm in conjunction with Latin hypercube sampling is developed to achieve computational efficiency. The convergence of the present algorithm for laminated composite curved panels with cutout is validated with original finite element (FE) analysis along with traditional Monte Carlo simulation (MCS). The variations of input parameters (both individual and combined cases) are studied to portray their relative effect on the output quantity of interest. The performance of the SVR based uncertainty quantification is found to be satisfactory in the domain of input variables in dealing low and high dimensional spaces. The layer-wise variability of geometric and material properties are included considering the effect of twist angle, cutout sizes and geometries (such as cylindrical, spherical, hyperbolic paraboloid and plate). The sensitivities of input parameters in terms of coefficient of variation are enumerated to project the relative importance of different random inputs on natural frequencies. Subsequently, the noise induced effects on SVR based computational algorithm are presented to map the inevitable variability in practical field of applications.

Metamaterial Properties of Periodic Laminates

Article

Jul 2016
J MECH PHYS SOLIDS

Ankit Srivastava

In this paper we show that a 1-D phononic crystal (laminate) can exhibit metamaterial wave phenomena which are traditionally associated with 2- and 3-D crystals. Moreover, due to the absence of a length scale in 2 of its dimensions, it can outperform higher dimensional crystals on some measures. This includes allowing only negative refraction over large frequency ranges and serving as a near-omnidirectional high-pass filter up to a large frequency value. First we provide a theoretical discussion on the salient characteristics of the dispersion relation of a laminate and formulate the solution of an interface problem by the application of the normal mode decomposition technique. We present a methodology with which to induce a pure negative refraction in the laminate. As a corollary to our approach of negative refraction, we show how the laminate can be used to steer beams over large angles for small changes in the incident angles (beam steering). Furthermore, we clarify how the transmitted modes in the laminate can be switched on and off by varying the angle of the incident wave by a small amount. Finally, we show that the laminate can be used as a remarkably efficient high-pass frequency filter. An appropriately designed laminate will reflect all plane waves from quasi-static to a large frequency, incident at it from all angles except for a small set of near-normal incidences. This will be true even if the homogeneous medium is impedance matched with the laminate. Due to the similarities between SH waves and electromagnetic (EM) waves it is expected that some or all of these results may also apply to EM waves in a layered periodic dielectric.

Effective in-plane elastic moduli of quasi-random spatially irregular hexagonal lattices

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References

Stochastic mechanics of metamaterials