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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

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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)

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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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Engineered honeycomb lattice materials with high specific strength and stiffness along with the advantage of programmable direction-dependent mechanical tailorability are being increasingly adopted for various advanced multifunctional applications. To use these artificial microstructures with unprecedented mechanical properties in the design of different application-specific structures, it is essential to investigate the effective elastic moduli and their dependence on the microstructural geometry and the physics of deformation at the elementary level. While it is possible to have a wide range of effective mechanical properties based on their designed microstructural geometry, most of the recent advancements in this field lead to passive mechanical properties, meaning it is not possible to actively modulate the lattice-level properties after they are manufactured. Thus the on-demand control of mechanical properties is lacking, which is crucial for a range of multi-functional applications in advanced structural systems. To address this issue, we propose a new class of lattice materials wherein the beam-level multi-physical deformation behavior can be exploited as a function of external stimuli like magnetic field by considering hard magnetic soft (HMS) beams. More interestingly, effective property modulation at the lattice level would be contactless without the necessity of having a complex network of electrical circuits embedded within the microstructure. We have developed a semi-analytical model for the nonlinear effective elastic properties of such programmable lattice materials under large deformation, wherein the mechanical properties can be modulated in an expanded design space of microstructural geometry and magnetic field. The numerical results show that the effective properties can be actively modulated as a function of the magnetic field covering a wide range (including programmable state transition with on-demand positive and negative values), leading to the behavior of soft polymer to stiff metals in a single lattice microstructure according to operational demands.

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Full-text available

Jun 2023

Traditionally lattice materials are made of a network of beams in two and three dimensions with majority of the lattice volume being void space. Recently researchers have started exploring ways to exploit this void space for multi-physical property modulation of lattices such as global mechanical behaviour including different elastic moduli, wave propagation, vibration, impact and acoustic features. The elastic moduli are of crucial importance to ensure the structural viability of various multi-functional devices and systems where a space-filled lattice material could potentially be used. Here we develop closed-form analytical expressions for the effective elastic moduli of space- filled lattices based on an exact stiffness matrix approach coupled with the unit cell method, wherein transcendental shape functions are used to obtain exact solutions of the underlying differential equation. This can be viewed as an accurate multi-material based generalization of the classical formulae for elastic moduli of honeycombs. Numerical results show that the effective in-plane elastic moduli can increase by orders of magnitude with a relatively lower infill stiffness ($\sim$10\%).This gives an exceptional opportunity to engineer multi-material lattices with optimal specific stiffness along with characterizing the mechanical properties of a multitude of lattice-like artificial and naturally occurring structural forms with space filling.

Programmable multi-physical mechanics of mechanical metamaterials

Article

Full-text available

May 2023
MAT SCI ENG R

Mechanical metamaterials are engineered materials with unconventional mechanical behavior that originates from artificially programmed microstructures along with intrinsic material properties. With tremendous advancement in computational and manufacturing capabilities to realize complex microstructures over the last decade, the field of mechanical metamaterials has been attracting wide attention due to immense possibilities of achieving unprecedented multi-physical properties which are not attainable in naturally-occurring materials. One of the rapidly emerging trends in this field is to couple the mechanics of material behavior and the unit cell architecture with different other multi-physical aspects such as electrical or magnetic fields, and stimuli like temperature, light or chemical reactions to expand the scope of actively programming on-demand mechanical responses. In this article, we aim to abridge outcomes of the relevant literature concerning mechanical and multi-physical property modulation of metamaterials focusing on the emerging trend of bi-level design, and subsequently highlight the broad-spectrum potential of mechanical metamaterials in their critical engineering applications. The evolving trends, challenges and future roadmaps have been critically analyzed here involving the notions of real-time reconfigurability and functionality programming, 4D printing, nano-scale metamaterials, artificial intelligence and machine learning, multi-physical origami/kirigami, living matter, soft and conformal metamaterials, manufacturing complex microstructures, service-life effects and scalability.

Non-invariant elastic moduli of bi-level architected lattice materials through programmed domain discontinuity

Article

Full-text available

May 2023
MECH MATER

Effective elastic moduli of lattice-based materials are one of the most crucial parameters for the adoption of such artificial microstructures in advanced mechanical and structural systems as per various application-specific demands. In conventional naturally occurring materials, these elastic moduli remain invariant under tensile and compressive normal modes or clock-wise and anti-clock-wise shear modes. Here we introduce programmed domain discontinuities in the cell walls of the unit-cells of lattice metamaterials involving a bi-level microstructural design to achieve non-invariant elastic moduli under tensile and compressive normal modes or clock-wise and anti-clock-wise shear modes. More interestingly, such non-invariance can be realized in the linear small deformation regime and the elastic moduli can be tailored to have higher or lower value in any mode compared to the other depending on the placement and intensity of the discontinuities in a programmable paradigm. We have derived an efficient analytical framework for the effective elastic moduli of lattice materials taking into account the influence of domain discontinuity. The axial and shear deformations at the beam level are considered along with bending deformation in the proposed analytical expressions. The numerical results ascertain that the domain discontinuities, in conjunction with unit cell level geometric parameters, can impact the effective elastic constants significantly under different modes of far-field stresses. It is further revealed that the degree of auxeticity of such lattices can be programmed to have target values (including non-invariance under different modes of deformation) as a function of the intensity and location of domain discontinuity when axial and shear deformations are included at the beam level. Realization of the unusual non-invariant elastic moduli of bi-level architected lattice materials would lead to a range of technologically demanding niche applications where one mode of deformation requires more or less force to deform compared to the opposite mode. Besides being able to perform as a load-bearing component, the proposed metamaterial can be used as an integrated sensor for measuring the level of stress or strain in structures.

The mechanics and optimal design of micro-architectured stepped hexagonal lattices

Article

Mar 2023
COMPOS STRUCT

An FFT-based method for estimating the in-plane elastic properties of honeycomb considering geometric imperfections at large elastic deformation

Article

Apr 2023
THIN WALL STRUCT

The geometric imperfections in honeycomb cells inevitably arise during the manufacturing process and can affect the honeycomb's mechanical performance. Hence, it is critical to identify and quantify the geometric imperfections and characterize the effects of these imperfections on the effective mechanical properties of honeycomb. In this study, the geometric imperfections inside the honeycomb were identified from micro-CT scan data. Using the obtained geometric topology, a Fast Fourier Transform (FFT) based method was employed in combination with the composite pixel method to predict the effective in-plane elastic properties of honeycomb at large elastic deformations. The results of the nonlinear homogenized prediction were compared with the experimental data and the prediction values from analytical models to investigate the effects of geometric imperfections. It was found that the predictions from the FFT-based method agreed well with the experimental results. However, the predictions from the analytical models significantly overestimated or underestimated the elastic properties, indicating that the geometric imperfections inside the honeycomb significantly influence its in-plane elastic properties.

Additively manufactured cure tools for composites manufacture

Article

Full-text available

Jun 2023
INT J ADV MANUF TECH

This research presents a novel framework for the design of additively manufactured (AM) composite tooling for the manufacture of carbon fibre-reinforced plastic composites. Through the rigorous design and manufacture of 30 unique AM tools, the viability of a design for AM framework was evaluated through measuring the performance with respect to geometrical accuracy and thermal responsiveness, and simulating the tool specific stiffness. The AM components consisted of a thin layup facesheet, stiffened by a low density lattice geometry. These tools were successfully used to layup and cure small composite components. The tooling was highly thermally responsive, reaching above 93% of the applied oven heating rate and up to 17% faster heating rates compared to similar mass monolithic tools. The results indicate that thermal overshoot has a greater dependence on the lattice density while the heating rate was more sensitive to the facesheet thickness. Lattice densities of as little as 5% were manufactured and the best overall geometry was a graded gyroid lattice with thicker walls near the surface and thinner walls at the base, attached to a 0.7 mm thick facesheet. The outputs from this research can provide a new route to the design and manufacture of mould tools, which could have significant impacts in the composites sector with new, lighter, more energy efficient tooling.

Programmed out-of-plane curvature to enhance the multi-modal stiffness of bending-dominated composite lattices

Article

Full-text available

Dec 2022
AIAA J

Conventional bending-dominated lattices exhibit less specific stiffness compared to stretching-dominated lattices while showing high specific energy absorption capacity. This article aims to improve the specific stiffness of bending-dominated lattices by introducing elementary-level programmed curvature through a multi-level hierarchical framework. The influence of curvature in the elementary beams is investigated here on the effective in-plane and out-of-plane elastic properties of lattice materials. The beam-like cell walls with out-of-plane curvature are modeled based on 3D degenerated shell finite elements. Subsequently, the beam deflections are integrated with unit cell level mechanics in an efficient semi-analytical framework to obtain the lattice-level effective elastic moduli. The numerical results reveal that the effective in-plane elastic moduli of lattices with curved isotropic cell walls can be significantly improved without altering the lattice-level relative density, while the effective out-of-plane elastic properties reduce due to the introduction of curvature. To address this issue, we further propose laminated composite cell walls with out-of-plane curvature based on the 3D degenerated shell elements, which can lead to holistic improvements in the in-plane and out-of-plane effective elastic properties. The proposed curved composite lattice materials would enhance the specific stiffness of bending-dominated lattices to a significant extent, while maintaining their conventional multi-functional advantages.

Analytical homogenization for equivalent in-plane elastic moduli of honeycomb structures with stiffened joints

Article

Jun 2023
THIN WALL STRUCT

A modulated fingerprint assisted machine learning method for retrieving elastic moduli from resonant ultrasound spectroscopy

Article

Full-text available

Apr 2023

We used deep-learning-based models to automatically obtain elastic moduli from resonant ultrasound spectroscopy (RUS) spectra, which conventionally require user intervention of published analysis codes. By strategically converting theoretical RUS spectra into their modulated fingerprints and using them as a dataset to train neural network models, we obtained models that successfully predicted both elastic moduli from theoretical test spectra of an isotropic material and from a measured steel RUS spectrum with up to 9.6% missing resonances. We further trained modulated fingerprint-based models to resolve RUS spectra from yttrium–aluminum-garnet (YAG) ceramic samples with three elastic moduli. The resulting models were capable of retrieving all three elastic moduli from spectra with a maximum of 26% missing frequencies. In summary, our modulated fingerprint method is an efficient tool to transform raw spectroscopy data and train neural network models with high accuracy and resistance to spectra distortion.

Effective elastic properties of two dimensional multiplanar hexagonal nanostructures

Article

Full-text available

Dec 2016

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

Effective in-plane elastic properties of auxetic honeycombs with spatial irregularity