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Reinaldo Goncalves, Bruno; Romanoff, JaniSize-dependent modelling of elastic sandwich beams with prismatic cores

Published in:International Journal of Solids and Structures

DOI:10.1016/j.ijsolstr.2017.12.001

Published: 01/04/2018

Document VersionPublisher's PDF, also known as Version of record

Published under the following license:CC BY-NC-ND

Please cite the original version:Reinaldo Goncalves, B., & Romanoff, J. (2018). Size-dependent modelling of elastic sandwich beams withprismatic cores. International Journal of Solids and Structures, 136, 28-37.https://doi.org/10.1016/j.ijsolstr.2017.12.001

International Journal of Solids and Structures 136–137 (2018) 28–37

Contents lists available at ScienceDirect

International Journal of Solids and Structures

journal homepage: www.elsevier.com/locate/ijsolstr

Size-dependent modelling of elastic sandwich beams with prismatic

cores

Bruno Reinaldo Goncalves ∗, Jani Romanoff

Aalto University, Department of Mechanical Engineering, Aalto FI-0 0 076, Finland

a r t i c l e i n f o

Article history:

Received 17 August 2017

Revised 10 November 2017

Available online 8 December 2017

Keywords:

Size effect

Sandwich panels

Couple stress theory

Prismatic cores

a b s t r a c t

Sandwich panel strips with prismatic cores are modelled using the modified couple stress theory and

their elastic size-dependent bending behaviour investigated. Compatibility between the discrete sand-

wich and continuum beam kinematics is first discussed. A micromechanics-based framework to estimate

effective mechanical properties is provided and unit cell models constructed with elementary beam el-

ements to determine the stiffness parameters of various prismatic cores. Numerical studies show that

the modified couple stress Timoshenko beam enhances the static deflection predictions of the classical

Timoshenko model. A sensitivity measure based on structural ratios is proposed to estimate the influence

of size effects in the global beam-level response. The parameters governing size effects in elastic sand-

wich beams are identified: Face-to-core thickness ratio, core density and topology, vertical corrugation

order and set of load and boundary conditions. Size effects are shown more pronounced in low-density

cores that rely on corrugation bending as shear-carrying mechanism. Based on the external load, bound-

ary conditions and sensitivity factor, one can assess whether size effects are non-negligible in a given

engineering structure.

© 2017 The Authors. Published by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license.

( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

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1. Introduction

All-metal prismatic sandwich panels are a lightweight alterna-

tive to conventional plates used in civil, mechanical, vehicle engi-

neering and other applications. The structure is composed of two

skins separated by a low-density, visibly discrete core, which gov-

erns the mechanical properties and can be tailored based on the

application requirements. In the marine industry, sandwich pan-

els with various cores have been studied to substitute the conven-

tional plate with ribs arrangement ( Kujala and Klanac, 2005 ). Pris-

matics such as the X- and Y-cores have been used for the double

hull design due to their relative simple production and impact per-

formance ( Naar et al., 2002; Ehlers et al., 2012 ). Advances in manu-

facturing encouraged the development of micro-architectured pris-

matics, whose multifunctional capacity includes superior heat and

energy absorption performances ( Valdevit et al., 2004; Gu et al.,

2001; Evans et al., 2001 ). Overall, the discrete character of pris-

matic cores leads to computational modelling challenges due to

the presence of consecutive length scales, as representing their ac-

tual geometry is usually inefficient. Effective continuum models are

∗ Corresponding author.

E-mail address: bruno.reinaldogoncalves@aalto.fi (B. Reinaldo Goncalves).

a

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https://doi.org/10.1016/j.ijsolstr.2017.12.001

0020-7683/© 2017 The Authors. Published by Elsevier Ltd. This is an open access article u

herefore sought to facilitate the analysis and optimization of such

tructures.

In a broad sense, continuum models of sandwich structures

re divided into ply-level layerwise and equivalent single layer

ESL) categories; see Carrera and Brischetto (2009) for details. In

he context of low-complexity single-layer models, the sandwich

heory (e.g. Allen, 1969; Zenkert, 1995 ) is the classical reference,

here deflections are decomposed into bending and shear terms

nd compatibility between faces and core is invoked. More re-

ently, the homogenization theory based on asymptotic expan-

ion of cell quantities has been used to describe the response

f periodic structures such as sandwich panels ( Hassani and Hin-

on, 1998; Buannic et al., 2003 ) based on classical and first-order

hear deformation theories. Prismatic sandwich panels compose

n application case where relatively flexible cores result in non-

egligible local effects and size dependency. An enhanced model is

herefore needed to include the internal cell stiffness to the contin-

um, improving the shear stress distribution description near dis-

ontinuities and capturing size effects.

In recent years, microstructure-dependent theories re-emerged

s an alternative to account for the influence of different scales

n the global structural response. Liu and Su (2009) have pro-

osed using a two-dimensional couple stress model and the-

nder the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37 29

Fig. 1. Global and local scales of the continuum model with adopted conventions

indicated.

o

f

Y

2

t

a

t

R

o

o

w

l

w

s

m

g

i

t

a

m

s

a

f

b

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m

e

p

a

e

b

o

o

2

2

l

i

o

i

t

e

s

t

v

s

o

Fig. 2. (a) Constant shear angle definition in the classical first-order continuum and

internal cell fluctuations (b) local rotation measure as the variation of shear angle

between cell boundaries ∂�− and ∂�+ .

Fig. 3. Kinematics of deformation of the modified couple stress Timoshenko beam.

w

w

o

v

a

e

s

c

f

i

c

b

d

i

s

b

b

i

c

t

t

w

d

2

i

b

U

w

r

e

s

i

i

ε

retically derived length scale parameters to capture size ef-

ects in cellular solids. The modified couple stress theory of

ang et al. (2002) and generalizations to beams (e.g. Park and Gao,

006; Ma et al., 2008; Reddy, 2011 ) and plates have received at-

ention due to its relative simplicity in parameter characterization

nd interpretation of higher-order terms, unlike earlier works on

he topic. It has been shown in Romanoff and Reddy (2014) and

omanoff et al. (2015) that the modified couple stress beam the-

ry as given in Reddy (2011) satisfactorily predicts the response

f web-core sandwich beams with semi-rigid joints. The concepts

ere extended by Reinaldo Goncalves et al. (2017) to study buck-

ing and free vibrations of transversely flexible sandwich beams

ith web- and homogeneous cores. The results obtained in these

tudies indicate that the couple stress theory has potential to

odel size effects, while retaining simplicity and requiring a sin-

le additional parameter that can be determined with sound phys-

cal meaning. Yet, the parameters governing the size effect magni-

ude and relevance in practical modelling of sandwich structures

re still unknown.

The objective of this work is to define an enhanced continuum

odel for sandwich beams with prismatic cores and study their

ize-dependent elastic bending response. The main assumptions

re followed by a description of the global scale beam, which is

ounded on the modified couple stress theory. A micromechanics-

ased framework to determine effective stiffness properties is then

hown. Unit cell models constructed with elementary beam ele-

ents and analyzed using the direct stiffness method are used to

stimate stiffness properties of various prismatic cores. The cou-

le stress beam is then used to investigate size effects, which are

lso validated against corresponding discrete finite element mod-

ls. In particular, structural ratios that govern the size-dependent

ehaviour are identified, and the effect of cell density, core topol-

gy, vertical corrugation order, face-to-core thickness ratio and set

f load and boundary conditions studied.

. Size-dependent sandwich continuum

.1. Modelling assumptions

Consider sandwich beams composed of n periodic cells of

ength s to a total length L . Continuum modelling requires the

dentification of two interacting scales, global and local, with co-

rdinates ( x, z ) and ( x l , z l ) respectively, x = ( i + 1/2) s + x l , where

= 0,1.. n is the unit cell index ( Fig. 1 ). The cells have one plane of

opological symmetry whose normal is aligned with the x -axis. An

quivalent continuum based on the Timoshenko beam theory as-

umes that the cell average shear deformation can be described in

he global scale through a constant shear angle with horizontal and

ertical components, γ xz = γ h + γ v , located at the reference point,

ee Fig. 2 (a). This implies that the cell internal shear fluctuations

f order higher than one must have the form of an odd function,

hich in deflections corresponds to ( Fig. 2 (a))

l (x l )

= − w

l (−x l

)+ 2 γh x

l in � (1)

If Eq. (1) is fulfilled, the derivatives are alternating even and

dd functions of the coordinate x l , such that the shear-induced cur-

ature and stresses also have zero average by definition; the equiv-

lent continuum is then energetically consistent. Sandwich pan-

ls with prismatic cores have the condition (1) violated near local

hear discontinuities such as concentrated loads or slope boundary

onditions. The shear response becomes a combination of periodic

rame-like and local bending modes, being a function of beam and

nternal cell parameters. Once this effect is sufficiently large, ex-

luding the localized effects from the global response might not

e a satisfactory assumption.

To account for the non-periodic shear stress distribution near

iscontinuities, it is necessary to introduce the unit cell stiffness

n the continuum model. A local rotation measure based on the

econd displacement gradient is used here ( Eq. 3 ), which can

e thought as the shear angle variation between consecutive cell

oundaries ∂�− and ∂�+ as additional kinematical compatibil-

ty between discrete and continuum ( Fig. 2 (b)). That way, the unit

ell is no longer represented by a point only, but also by the varia-

ion between two consecutive points. Assuming a constant cell ro-

ation implies homogeneous cell internal stiffness, rendering piece-

ise constant local stresses decaying asymptotically away from the

iscontinuity point.

.2. Couple stress continuum model

Consider the modified couple stress Timoshenko beam model

n Ma et al. (2008) and Reddy (2011) . The displacement field can

e expressed in the form

x ( x, z ) = zφx ( x ) , U z ( x, z ) = w ( x ) (2)

here φx ( x ) is the transverse rotation of the cross-section at the

eference axis and w ( x ) the transverse displacement of the refer-

nce axis ( Fig. 3 ). As the bending response is the only object of

tudy, the mid-plane axial displacement and consequently the ax-

al stress resultant are not included in the model. The strain field

s given by ( Reddy, 2011 )

xx = z ∂ φx

∂x , γxz = φx +

∂w

∂x , χxy =

1

4

(∂ φx

∂x − ∂ 2 w

∂ x 2

)(3)

30 B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37

Table 1

Stiffness parameters for sandwich panels with selected prismatic cores and n c = 1.

Web-core ( Fig. 5 (a)) D Q E

( 1 −υ2 )

2 t 3 c t 3 f

s ( 2 dt 3 f + st 3 c )

S xy 2 3

E ( 1 −υ2 )

t 3 f

Triangular corrugated core ( Fig. 5 (b)) D Q E

( 1 −υ2 )

4 d 2 s t f t c 8 p 3 t f + s 3 t c

S xy 2 3

E ( 1 −υ2 )

( t 3 f

+

s 4 p

t 3 c )

Hexagonal honeycomb core, n c = 1 ( Fig. 5 (c)) D Q E

( 1 −υ2 )

432 t 3 f t 3 c ( 2 t

3 f + t 3 c )

s 2 ( 63 t 6 f +48 t 3

f t 3 c +8 t 6 c )

S xy E

( 1 −υ2 )

2 t 3 f ( 2 t 3 c + t 3 f )

3 t 3 f +4 t 3 c

Y-core ( Fig. 5 (d)) D Q Et 3 c

( 1 −υ2 ) s ( d

h ) 2

3 p( 8 h + p ) t 6 f +4 s ( 3 h + p ) t 3

f t 3 c + s 2 t 6 c

12 hp( 2 h + p ) t 6 f + s ( 12 h 2 +16 hp+3 p 2 ) t 3

f t 3 c +2( 2 h + p ) s 2 t 6 c

S xy 2 3

E ( 1 −υ2 )

( t 3 f

+

s 4 p

t 3 c )

Fig. 4. Unit cell models for equivalent stiffness determination (a) bending stiffness

(b) transverse shear stiffness (c) local bending stiffness.

a

u

w

w

d

D

2

h

Q

N

w

a

T

b

c

b

c

r

s

γ

w

n

where εxx and γ xz are the normal and shear strains, whereas χ xy

is the local cell curvature. The bending, shear and local bending

stress resultants are respectively ( Reddy, 2011 )

M xx = D xx ∂ φx

∂x , Q x = D Q

(φx +

∂w

∂x

), P xy =

S xy

2

(∂ φx

∂x − ∂ 2 w

∂ x 2

)

(4)

Likewise, D xx , D Q and S xy represent the bending, transverse

shear and local bending stiffness terms, which are defined for pris-

matic sandwich panels through unit-cell analysis (see Section 2.3 ).

The static equilibrium equations for the homogeneous beam under

a transverse distributed load q is given by ( Reddy, 2011 )

∂ Q x

∂x +

1

2

∂ 2 P xy

∂ x 2 = −q, Q x − ∂ M xx

∂x − 1

2

∂ P xy

∂x = 0 (5)

and the boundary conditions ( Reddy, 2011 )

Q x +

1

2

∂ P xy

∂x or w, M xx +

1

2

P xy or φx , 1

2

P xy or θx = −∂w

∂x (6)

The underlying static equilibrium equations may be solved an-

alytically, for instance using the Navier procedure ( Reddy, 2011 ),

or numerically, with the finite element method as the most com-

mon choice ( Arbind and Reddy, 2013; Reinaldo Goncalves et al.,

2017 ). Alternatively, the exact solution can be employed; closed-

form equations for common loading cases have been derived in

Karttunen et al. (2016) . The classical Timoshenko beam theory is

recovered by constraining the local rotation; meanwhile, the mod-

ified couple stress Euler–Bernoulli model is obtained if φx = θ x .

2.3. Unit cell models and micromechanics-based effective properties

Consider a two-dimensional unit cell i = n of an arbitrary pris-

matic sandwich beam of unit thickness in the local coordinate sys-

tem (see Fig. 1 ). The cell body is denoted �, with origin located at

mid-length and vertical neutral axis. The boundaries ∂�± defined

from topological periodicity are subjected to boundary tractions,

while top and bottom boundaries ∂�t,b are traction-free. Three cell

boundary value problems, where equivalent M xx , Q x , P xy ( Eq. (4 ))

are enforced one at a time while the others are set to zero, are

used to compute effective properties. The average cell stresses and

strains are determined using the principles of cell micromechanics

( Sun and Vaidya, 1996; Lok and Cheng, 20 0 0 ). The unit cell models

used for the computations and their underlying boundary condi-

tions are shown next. In addition to the boundary conditions dis-

cussed, the cells must be further restrained at the horizontal sym-

metry plane to prevent rigid body motion.

2.3.1. Bending stiffness D xx

Fig. 4 (a) shows a unit cell under horizontal forces in oppo-

site directions applied at top and bottom faces, i.e. N = N t = −N

b D

t x l = −s /2. The following boundary conditions are proposed

l (−s/ 2 , z l

)= −u

l (s/ 2 , z l

)

l (−s/ 2 , z l

)= w

l (s/ 2 , z l

) (7)

The average bending curvature is determined

∂φx

∂x =

2 δh

sd (8)

here δh = u l ( s /2, d /2) − u l ( s /2, −d /2). The bending stiffness re-

uces to

xx =

Ns d 2

2 δh

(9)

.3.2. Transverse shear stiffness D Q

Fig. 4 (b) shows a unit cell under the transverse force Q x and

orizontal forces N required for static equilibrium

x = V t + V b + V c i

t = −N b =

Q x s 2 d

(10)

here V t , V b and V c i are the transverse force at top, bottom faces

nd i th core member across the unit cell boundary respectively.

he displacement boundary conditions are proposed

w

l (s/ 2 , z l

)= −w

l (−s/ 2 , z l

)= δv

u

l (s/ 2 , z l

)= u

l (−s/ 2 , z l

)= δh

(z l ) (11)

Based on Eqs. (10 )–(11) , it is implied that core boundary mem-

ers can deform horizontally in periodic fashion (see Fig. 2 (a)). The

ell does not necessarily deform as a parallelogram, which would

e an overly stiff requirement ( Sun and Vaidya, 1996 ). The verti-

al boundary displacement δv is constant to ensure that the cell

emains incompressible after deformation. The average cell shear

train is then obtained

xz =

2 δv

s +

�δh

d (12)

here �δh = u l (x l , d /2) − u l (x l , − d /2). The transverse shear stiff-

ess reduces to

Q =

Q x sd (13)

2 δv d + �δh s

B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37 31

Fig. 5. Unit cell models of prismatic sandwich beams with single order of corruga-

tion ( n c = 1), (a) web-core (b) triangular corrugated core (c) hexagonal honeycomb

core (d) Y-core.

Fig. 6. Unit cell models of (a) diamond core (b) hexagonal honeycomb core sand-

wich beams with various orders of corrugation.

Table 2

Size-effect sensitivity parameter for sandwich panels with selected

prismatic cores; face-to-core thickness ratio k t = t f / t c .

Size-effect sensitivity parameter ∇

2

Web-core 1 3 ( s

L ) 2 ( 2 d

s k 3 t + 1 )

Triangular core 1 12

( s L ) 2 ( t c

d ) 2

( 2 k 3 t + cos α)( cos α3 + k t ) k t cos α3

X-core ( Diamond, n = 2) 1 3 ( s

L ) 2 ( t c

d ) 2

k 3 t + cos αcos α3

Hexagonal core ( n = 1) ( s L ) 2

( k 3 t +2 )( 63 k 6 t +48 k 3 t +8 )

864( 2 k 3 t +1 )( 3 k 3 t +4 )

Y-core ( h = d/ 2 = s /2) 1 24

( s L ) 2

( 4 k 3 t + √

2 )[ 6( 1+ √

2 ) k 6 t +( 9+8 √

2 ) k 3 t +2 √

2 +4 ]

3( 1+4 √

2 ) k 6 t +4( 3+ √

2 ) k 3 t +2

2

b

n

g

l

d

c

r

P

θ

b

l

χ

a

S

2

c

c

a

l

s

t

a

t

s

t

r

g

D

w

s

w

2

b

D

S

a

3

r

n

o

a

p

t

m

n

h

m

p

I

f

t

∇

s

i

b

2

w

i

o

t

p

.3.3. Local bending stiffness S xy

Unlike the previous quantities, the deformation mode induced

y P xy is not periodic in �. Yet, by assuming that the local stiff-

ess varies slowly in �, that is, that the cell is relatively homo-

eneous in terms of its internal constituents, an estimate of the

ocal stiffness can be obtained. Consider the cell of Fig. 4 (c) under

istributed edge bending moments. In this mode, the internal cell

onstituents are flexural only, such that P xy induces purely internal

otations. The couple-stress resultant in Eq. (4) is defined

xy = 2 ( M t + M b + M c i ) (14)

The following slope boundary condition is proposed

x

(s/ 2 , z l

)= −θx

(−s/ 2 , z l

)= θ l (15)

Assuming that the local rotation is much higher than the whole

eam curvature as expected in size-effect sensitive structures, the

ocal rotation vector reduces to

xy =

1

4

(2 θ l

s

)=

θ l

2 s (16)

nd the local bending stiffness is obtained

=

P xy s (17)

xy

θ l T

.4. Elastic constants for selected prismatic cores

Stiffness parameters are derived for selected prismatic cores ac-

ording to the micromechanical models shown in Section 2.3 . The

ells are constructed using Euler–Bernoulli beam elements and an-

lyzed using the direct stiffness method. For the derivation of the

ocal bending stiffness S xy , the axial degree of freedom is con-

trained, thus the elements are purely flexural. Face and core ma-

erial properties are the same, defined by the elastic modulus E

nd Poisson’s ratio ν . Top and bottom faces have equal thickness

f in all cases. It is assumed in the derivations that the cell con-

tituents are thin-walled, that is, t f , t c � s, d, h, p . The corruga-

ions are considered perfectly connected to the faces, composing

igid assemblies with combined properties. Based on Eq. (9) , the

lobal bending stiffness is approximated in all cases by

xx ≈ E

1 − υ2

t f d 2

2

(18)

Equivalent shear and local bending stiffness properties for sand-

ich beams with single order of corrugation, n c = 1, ( Fig. 5 ) are

ummarized in Table 1 .

Fig. 6 shows the unit cells of multi-layered sandwich panels

ith diamond and hexagonal honeycomb cores (e.g. Valdevit et al.,

004; Gu et al., 2001 ). The stiffness parameters for diamond core

eams ( Fig. 6 (a)) is given in the compact form

Q =

E (1 − υ2

) 1

2 n c

s d 2 t c

p 3 (19)

xy =

2

3

E (1 − υ2

)(t 3 f +

n c s

4 p t 3 c

)(20)

Stiffness parameters for hexagonal honeycomb core ( Fig. 6 (b))

re provided in Appendix A for selected corrugation orders.

. Size-dependent sandwich response

Stiffness size dependency is observed in sandwich panels as

esult of the relative influence of non-periodic stress distribution

ear discontinuities in the global structural response. The effect

f discontinuities, such as concentrated loads and slope bound-

ry conditions, vanishes asymptotically away from their application

oint and may be relevant if this range is non-negligible in relation

o the overall beam length. Classical continuum models with strain

easure based on the first displacement gradient have no mecha-

ism to include the cell structural length scale in the analysis, and

ence predict size-independent response. Conversely, single-layer

odels that include higher-order displacement gradients can sup-

lement the continuum with information from the unit cell scale.

n the modified couple stress Timoshenko beam (MCSTBT), size ef-

ects arising from the local scale stiffness are estimated through

he non-dimensional relation

=

√

S xy

D Q L 2 (21)

If face and core material properties are equal, the magnitude of

uch size effects for a given set of loading and boundary conditions

s uniquely dependent on the cell geometric parameters and length

etween supports L . For selected prismatic geometries in Section

.4 ( Figs. 5 and 6 ) the resulting ∇

2 are given in Table 2 .

To investigate size effects in the continuum modelling of sand-

ich beams with prismatic cores, selected topologies are studied

n the linear elastic range. The responses are compared to the

nes obtained with the classical Timoshenko beam theory (TBT)

hroughout the study. To emphasize the size effect, deflections are

resented relative to the maximum deflection obtained with the

BT. In all analyses, the material properties E = 206 GPa and ν = 0.3

32 B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37

Fig. 7. Example of three-dimensional finite element model used for validation throughout this work (web-core panel strip in three-point bending and mid-length symmetry).

Fig. 8. Discrete cell geometry and deflection lines obtained with classical and non-classical continua and discrete finite elements.

3

d

e

t

t

w

are assumed (steel). Discrete three-dimensional finite element (3D

FE) models are used for validation ( Fig. 7 ). The validation models

represent the actual panel geometry, finely discretized with Abaqus

S4R shell elements ( Dassault Systemes, 2013 ). Boundary conditions

are applied as to ensure transverse incompressibility at the cross-

sections under loading and supports. At the surfaces perpendicular

to the beam main axis, plane strain conditions guarantee beam-like

response.

A

s

.1. Discrete response vs. classical and non-classical continuum

escriptions

Consider prismatic sandwich panel strips with unit-cell prop-

rties shown in Fig. 8 . The lengthwise deflection distribution is

raced for limiting cases using classical and size-dependent con-

inuum theories. In all cases, the beams are in three-point bending,

ith total lengths L A = 2.4 m, L B = 0.64 m and L C = 0.72 m. The cases

-C are selected based on their shear-flexibility and sensitivity to

ize effects. In short, they can be summarized as follows,

B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37 33

Fig. 9. Maximum deflection predictions for web-core panel strip using beam and

reference models as function of (a) relative unit cell size s / L (b) unit cell aspect

ratio d / s (c) face-to-core thickness ratio t f / t c .

w

(

a

E

w

S

fl

a

m

s

d

M

t

s

i

C

d

b

f

t

m

3

s

m

t

m

g

o

t

d

s

3

s

b

a

f

3

d

d

r

b

3

d

l

i

i

a

3

l

T

s

ρ

w

s

A) Shear stiff, θQ = 0.0 0 05; low size-effect sensitivity ∇ = 0.0014;

B) Shear flexible θQ = 1.530; moderate size-effect sensitivity

∇ = 0.035;

C) Shear flexible θQ = 1.589; high size-effect sensitivity ∇ = 0.109;

here θQ = D xx / L 2 D Q measures the shear flexibility of the beam

Zenkert, 1995 ). In addition to the TBT and MCSTBT models, we

lso present the curves obtained with the modified couple stress

uler–Bernoulli (MCSEBT) beam theory ( Park and Gao, 2006 ),

herein the non-classical parameter is taken the same as in

ections 2.3 and 2.4 .

The MCSTBT model is observed to predict the average static de-

ection distributions accurately in all cases. In case A ( Fig. 8 (a)),

bending-dominated problem, the three models render approxi-

ately the same deflection lines. Case B ( Fig. 8 (b)) shows a shear-

ensitive problem; the MCSEBT has no mechanism to include shear

eformations, thus predicts overly stiff response. The TBT and

CSTBT models behave similarly, and a kink is observed under

he concentrated force as the unit cell has little internal bending

tiffness. Local bending is observed only in the immediate vicin-

ty of the concentrated force and its effect is nearly negligible.

ase C ( Fig. 8 (c)) emphasizes the need of the MCSTBT for a size-

ependent continuum; the unit cell stiffness is high, thus local

ending is non-negligible in the entire cell near the concentrated

orce. This behaviour cannot be described with TBT and MCSEBT

heories due to the lack of length-scale parameter and shear defor-

ation measure respectively.

.2. Fundamental ratios governing the size effect: web-core case

tudy

Table 2 indicates that size effect in sandwich panels with pris-

atic cores is governed by certain structural ratios. To understand

heir influence in the response prediction with TBT and MCSTBT

odels, we select a simple geometry with weak coupling among

eometric parameters. Consider a web-core sandwich panel strip

f unit width in three-point bending. For this geometry, three ra-

ios govern the size effect, which are studied here separately: s / L,

/ s and t f / t c (see Fig. 5 (a)). The sandwich panel depth is taken con-

tant, d = 0.05 m for all analyses. In (a) and (b), t f = 0.004 m.

.2.1. Relative unit cell size s / L ( Fig. 9 (a))

Let d / s = 5/8 and t f / t c = 1.6 be constant and the relative unit cell

ize 1/36 ≤ s / L ≤ 1/6. As predicted in Table 2 , the local effects

ecome negligible as s / L → 0, hence the TBT model is incrementally

ccurate. The relative difference between beam models is a linear

unction of s / L , which accounts for the local effects magnitude.

.2.2. Unit cell aspect ratio d / s ( Fig. 9 (b))

In this analysis, t f / t c = 1 and s = L /6 are constant, while 1/10 ≤ / s ≤ 2. The parameter ∇ is here proportional to

√

d/s + 1 / 2 . As

/ s → 0, the local effects are no longer dependent on the cell aspect

atio, but still on the other ratios; the relative difference between

eam models becomes therefore nearly constant.

.2.3. Face-to-core thickness ratio t f / t c ( Fig. 9 (c))

To study the role of member thicknesses, we select s = L /6 and

/ s = 5/8, with 1/8 ≤ t f / t c ≤ 2. It is shown that the magnitude of

ocal effects does not depend substantially on this ratio if t f < t c ,

n fact, converging to a constant as t f / t c → 0. Conversely, if t f > t c ,

ncrementally increasing the ratio results in ∇ rapidly increasing

s a function of ( t f / t c ) 3/2 .

.3. Role of core topology and relative density

The position of corrugations and their relative density in re-

ation to the overall unit cell area affect the structural ratios of

able 2 and thus the size-effect sensitivity. The core relative den-

ity is defined in the general case

c =

1

sd

j ∑

i =1

t c l c,i (22)

here l c,i is the length of the i th to a total of j core members. We

tudy common prismatic cores with single and multiple orders of

34 B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37

Fig. 10. Core geometries and dimensions used for the core density analysis, n c = 1.

Fig. 11. Core geometries and dimensions for panels with varying corrugation

order n c .

u

t

c

a

i

o

t

o

3

f

n

i

t

n

i

d

e

c

d

i

t

e

l

g

c

e

c

f

l

4

o

s

U

w

r

t

M

s

I

v

b

m

i

l

o

s

i

g

corrugation to compare their behaviour and validate the MCSTBT

in terms of maximum deflection against three-dimensional finite

element models. Face thicknesses and depths are scaled to obtain

constant bending stiffness ( D xx = 848.9 kNm), while the thickness

of the corrugations is adjusted as function of the core relative den-

sity.

3.3.1. Common prismatic cores with n c = 1

Consider prismatic sandwich panels with dimensions shown in

Fig. 10 . Fig. 12 (a) shows the parameter ∇ ( Table 2 ) in terms of ρc

for the geometries selected. An association of size-effect sensitiv-

ity with density and shear-carrying mechanism is observed. Cores

whose main mechanism involves bending of the corrugations dis-

play dramatic increases in sensitivity as their slenderness is re-

duced. In contrast, cores that rely on membrane stretching have

lower sensitivity overall, which only marginally depends on ρc .

Fig. 13 shows the maximum deflection predicted by the mod-

els as function of the core density, 0.01 ≤ ρc ≤ 0.12, for a unit-

width panel with L = 0.8 m in three-point bending. In size-effect

sensitive cores, the TBT exhibits overly soft response as the den-

sity is reduced. In contrast, the MCSTBT is able to predict deflec-

tions very satisfactorily for all cases studied with maximum rela-

tive error under 2%. Differences between MCSTBT and 3D FE are

attributed to internal cell fluctuations due to higher-order deriva-

tives of displacement, which are averaged to a constant curvature

as discussed in Section 2 .

3.3.2. Multi-layered prismatic cores

Consider unit-width panels with hexagonal honeycomb and di-

amond cores and varying core volume ratio as shown in Fig. 11 .

Fig. 12 (b) shows the sensitivity parameter ∇ as function of the cor-

rugation order and core relative density. In hexagonal honeycomb

panels, the sensitivity increases with the corrugation order in low-

density cores. The increments are increasingly smaller as n c grows,

tending to a constant as the cross-section becomes homogeneous,

n c → ∞ . In contrast, in diamond core panels, the corrugation order

plays a minimal role in the panel stiffness and size-effect sensitiv-

ity. The behaviour is consistent with the shear-carrying mechanism

dominant in each core: bending of cell walls in hexagonal honey-

combs and corrugation stretching in diamond cores.

Fig. 14 validates the MCSTBT against three-dimensional finite

elements in terms of maximum deflections. The analyses are un-

dertaken for a doubly-clamped unit-width 0.4 m long panel strip

nder centred vertical force. It is shown that the maximum deflec-

ion increases with the corrugation order in hexagonal honeycomb

ores due to the increased shear-flexibility as the wall thicknesses

re reduced ( Fig. 14 (a)). The increase is considerably more subtle

n diamond core panels ( Fig. 14 (b)). It is shown that the response

f multi-layered sandwich panels can be accurately predicted with

he MCSTBT regardless of their geometry, density or corrugation

rder.

.4. Role of boundary conditions

Besides the sensitivity parameter, the magnitude of size ef-

ects in prismatic sandwich structures depends on the loading sce-

ario, as concentrated loads and restrictive boundary conditions

nduce the size-dependent behaviour. In the context of concen-

rated forces, it is of particular interest how severe is the disconti-

uity in the shear force diagram. We compare three common load-

ng cases in which size effect is observable: simply supported and

oubly clamped beams under mid-span concentrated force and

nd-loaded cantilever.

Consider panels with web-core, Y-core and hexagonal honey-

omb core ( n c = 1) and dimensions shown in Fig. 10 . The relative

ifference in maximum deflections obtained with TBT and MCSTBT

s shown in Fig. 15 as function of ∇ . The difference highlights

he local effects associated with the second displacement gradi-

nt and average cell stiffness. Overall, the relative difference is a

inear function of ∇ when local effects are moderate, with slope

overned by the discontinuity type. It is largest for the doubly

lamped beam, as multiple discontinuities are present, and low-

st for the cantilever beam, where the only discontinuity is the

lamped end. The results of previous sections can be readily scaled

or other load conditions, with the size effect becoming more or

ess severe.

. Discussion and conclusions

In this study, the modified couple stress Timoshenko beam the-

ry ( Ma et al., 2008; Reddy, 2011 ) has been applied to study the

ize-dependent response of sandwich beams with prismatic cores.

nlike previous works ( Dai and Zhang, 2008; Liu and Su, 2009 ),

e consider solely local stiffening causing disturbances in the pe-

iodic shear field as size effect, as changes in flexural rigidity due

o material positioning can be modelled using classical continua.

oreover, the size effects considered are only due to the length-

cale interactions, and not to other local shear-field disturbances.

t has been assumed that the structures are a repetition of trans-

ersely rigid complete cells. In reality, prismatic structures might

e transversely flexible ( Frostig and Baruch, 1990 ) and the edges

ay contain incomplete cells ( Anderson and Lakes, 1994 ) promot-

ng other types of size effects, whose influence in the response of

ightweight sandwich structures is left for future work.

In this study, we have chosen the modified couple stress Tim-

shenko beam theory (MCSTBT) given its simplicity; it includes a

ingle non-classical stiffness parameter in its formulation, whose

nterpretation in the context of prismatic sandwich panels is tan-

ible. A downside is that the local stiffness must be considered

B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37 35

Fig. 12. Size-effect sensitivity parameter as function of the core geometry and relative density ratio (a) cores with order of corrugation n c = 1 (b) hexagonal and diamond

cores with varying n c .

Fig. 13. Maximum deflection predictions for (a) web-core (b) triangular corrugated core (c) hexagonal honeycomb core (d) Y-core sandwich panels with varying core relative

volume ratio using beam and validation models.

c

s

e

m

i

(

w

w

s

o

d

o

l

onstant, while in reality the cells are highly discrete. Further con-

ecutive scales could be introduced through higher-order gradi-

nts of displacement improving the cell stiffness description. The

icromechanics-based stiffness determination framework general-

zes the works of Romanoff and Reddy (2014) and Romanoff et al.

2015) in studying web-core panels with semi-rigid joints; in their

ork, equivalence with the classical sandwich theory ( Allen, 1969 )

as invoked to determine the non-classical stiffness. It has been

hown that the modified couple stress Euler–Bernoulli beam the-

ry ( Park and Gao, 2006 ) is not suitable to model the size-

ependent behaviour of these structures, as the size effects arise

nly when the shear flexibility is high.

Size effects in prismatic sandwich beams are shown to be re-

ated to certain structural ratios involving cell and overall beam

36 B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37

Fig. 14. Maximum deflection predictions for doubly-clamped (a) hexagonal honeycomb core (b) diamond core sandwich panels with two orders of corrugation as function

of the core relative density.

Fig. 15. Relative difference between TBT and MCSTBT models for different sets of

loading and boundary conditions as function of the parameter ∇ .

Table A.1

Transverse shear stiffness D Q and local bending stiffness S xy of hexagonal

honeycomb core beams with selected orders of corrugation n c .

n c D Q S xy

2 E ( 1 −υ2 )

54 t 3 c ( 281 t 6 f +255 t 3

f t 3 c +8 t 6 c )

s 2 ( 693 t 6 f +708 t 3

f t 3 c +88 t 6 c )

E ( 1 −υ2 )

14 t 6 f +15 t 3

f t 3 c +2 t 6 c

21 t 3 f +5 t 3 c

3 E ( 1 −υ2 )

432 t 3 c ( 668 t 6 f +788 t 3

f t 3 c +31 t 6 c )

s 2 ( 9398 t 6 f +11801 t 3

f t 3 c +1200 t 6 c )

E ( 1 −υ2 )

52 t 6 f +82 t 3

f t 3 c +14 t 6 c

78 t 3 f +19 t 3 c

5 E ( 1 −υ2 )

10800 t 3 c ( 273242 t 6 f +391311 t 3

f t 3 c +17223 t 6 c )

s 2 ( 60753591 t 6 f +89833161 t 3

f t 3 c +7281992 t 6 c )

E ( 1 −υ2 )

724 t 6 f +1866 t 3

f t 3 c +372 t 6 c

1086 t 3 f +265 t 3 c

A

c

p

s

R

A

A

A

B

C

D

E

F

G

H

dimensions, being the exponent of each ratio a measure of its sen-

sitivity. Such ratios have been studied in an example involving a

simple geometry, for which the effects are easily isolated. The in-

fluence of core density and topology in the size-dependency has

been then investigated. Overall, it has been shown that size ef-

fect is more pronounced in cores where corrugation bending is the

dominating shear-carrying mechanism. The core relative density

governs the size effect magnitude, which is also influenced by the

corrugation order in certain prismatic panels such as the hexag-

onal honeycomb type. Lastly, the influence of loading scenario in

the size effect magnitude has been investigated; three simple cases

have been selected, and the doubly clamped under mid-length ver-

tical load seen to be critical as it introduces the most severe set of

discontinuities. While the material properties of faces and core are

taken for simplicity equal in this study, the size effect magnitude

should otherwise also dependent on their ratio.

Acknowledgements

The authors gratefully acknowledge the financial support from

the Graduate School of Aalto University School of Engineering and

the partners of the Finland Distinguished Professor programme

“Non-linear response of large, complex thin-walled structures”:

Tekes, Napa, SSAB, Deltamarin, Koneteknologiakeskus Turku and

Meyer Turku.

ppendix A. Stiffness parameters for hexagonal honeycomb

ore panel

The coefficients for the stiffness of hexagonal core sandwich

anels ( Fig. 6 (b)) with selected orders of corrugation n c are pre-

ented in Table A.1 .

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