Modeling of Interactive Buckling in Sandwich Struts with Functionally Graded Cores

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Journal of Engineering Mechanics, Yiatros et al. (2012)

Modelling of interactive buckling in sandwich struts with functionally graded cores

Stylianos Yiatros1, M. Ahmer Wadee2 and Christina Völlmecke3

Abstract:

An analytical pilot model for interactive buckling in sandwich struts with cores made from a Functionally

Graded Material (FGM) based on total potential energy principles is presented. Using a Timoshenko beam

approach, a system of nonlinear differential and integral equations is derived that predicts critical and secondary

instabilities. These are validated against numerical simulations performed within the commercial finite element

package ABAQUS. Good agreement is found and this offers encouragement for more elaborate models to be

devised that can account for face–core delamination – a feature where FGMs are known to offer distinct

advantages.

CE Database subject Headings: Structural stability, Interactive buckling, Post-buckling, Functionally Graded

Materials, Sandwich structures

1. Introduction

Sandwich construction comprising two stiff plates separated by a core of softer material boasts its

advantage over homogeneous construction methods due to its optimized configuration which assigns material

and stiffness where it is most effective, thus reducing the cost and weight of the structural element. Owing to

their merits, sandwich structures can be found in a wide range of engineering applications ranging from

astronautics to civil engineering (Zenkert, 1995). A problem with optimized structural forms, such as sandwich

struts, is that since these are tuned for a specific objective they can become prone to more than one instability

mode being triggered simultaneously. Specifically for the case of a sandwich strut under uniaxial compression,

1Lecturer, Department of Civil Engineering & Geomatics, Cyprus University of Technology, Limassol, Cyprus.Email: [email protected] Reader in Nonlinear Mechanics, Department of Civil & Environmental Engineering, Imperial College London, London, UK. Email: [email protected], LKM, Institut für Mechanik, Technische Universität Berlin, Germany. Email: [email protected]

Journal of Engineering Mechanics. Submitted January 31, 2012; accepted July 27, 2012; posted ahead of print August 2, 2012. doi:10.1061/(ASCE)EM.1943-7889.0000470

Copyright 2012 by the American Society of Civil Engineers

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after the mode with the lowest critical buckling load is triggered, the resulting deformations give rise to

increasing stresses that may trigger additional buckling modes through a secondary bifurcation point. Known as

interactive buckling (Fig. 1) this can lead to an unstable post-buckling response since the least stiff equilibrium

path is followed. Consequently, this can have significantly negative effects on the sandwich strut, causing it to

lose a considerable proportion of its load carrying capacity (Atankovic, 1997; Hunt & Wadee, 1998; Huang &

Kardomateas, 2002), while in the presence of imperfections the proximity of the critical and the secondary

bifurcations may even cause premature failure (Wadee, 2000; Beghini et al., 2006). Another major issue

affecting the load carrying capacity and structural integrity of sandwich structures is delamination or debonding

at the face plate-core interface and its propagation due to buckling (Østergaard, 2008). Delamination, usually

induced by impact or buckling, can be characterized as a great challenge in composites since it prevents the

composite action between the face plate and the core locally, while its propagation magnifies the problem with

detrimental consequences.

A promising development for the engineering industry is the use of Functionally Graded Materials

(FGMs) in sandwich construction. These materials have gained importance in astronautical technology where

significant reductions in cost can be achieved by utilizing materials that combine two or more functions safely,

resulting in a reduction of the payload (Zhu & Sankar, 2007). Sandwich construction with FGMs is mainly seen

in sandwiched FGM plates and sandwich struts with FGM face plates where the core is usually homogeneous.

This is the case in aerospace applications, where structural elements are subjected simultaneously to thermal

and mechanical loads; the sandwich struts are expected to perform well at elevated temperatures caused by

friction in the atmosphere during hypersonic flight. The thermo-elastic properties of the sandwich struts

therefore need to be controlled by gradually varying the ceramic and metallic composition of the face plates.

The metallic component, other than the strength, is expected to contribute to the toughness while the ceramic

component would also contribute to corrosion resistance (Zenkour, 2005). More related to the scope of work

herein are the applications of sandwich struts with functionally graded cores where the stiffness varies in the

transverse direction by gradually changing the density of the foam material. Grading the material properties of

the core can provide further advantages to sandwich construction, such as the increase of critical loads, but it

can also address specific localized phenomena such as delamination and fracture due to impact (Anderson,

2003).

In terms of the structural analysis of functionally graded sandwich struts, there is great focus on the

application of mechanical and thermal loads (Samsam Shariat et al., 2005; Zenkour, 2005; Zhu & Sankar,



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2007), with special attention to the change in the response brought by the gradation. It has been shown that the

risk of delamination due to high interfacial shear stresses is minimized if the stiffness of the weaker part is

locally increased (Venkataraman & Sankar, 2003; Anderson, 2003). The higher stiffness near the interface also

results in an increase of the wrinkling, or local buckling, critical load (Ávila, 2007) as well as helping to reduce

electric displacement intensity factors in piezoelectric FGM (Li & Weng, 2002). The accurate modelling of the

kinematics of sandwich FGMs has been of great interest to many researchers leading to the proposal of several

approaches from simpler Equivalent Single Layer (Reissner, 1945; Reddy, 1984) to Layer-wise (Cho et al.,

1991) and Carrera’s unified formulation (Carrera et al., 2008); these approaches listed here in order of

increasing complexity and computational cost. Recently, Apetre et al. (2008) compared two Equivalent Single

Layer approaches with a Higher Order theory (Frostig et al., 1992) and a Fourier–Galerkin theory (Zhu &

Sankar, 2007) showing that the latter two approaches correlate much better with an FE simulation used for

validation, at the expense of increased computational cost. The reason lies in the fact that both approaches rely

on the solution of the elasticity equations without making any assumptions about the through-depth in-plane

displacements, unlike the Equivalent Single Layer approaches which are based on the same assumptions as the

analytical model presented herein. On similar grounds, Brischetto (2009) also suggests the use of higher order

theories in the cases of thick plates or large gradients.

The stability of FGM has also been investigated in terms of compression (Feldman & Abudi, 1997)

and thermal buckling (Samsam Shariat & Eslami, 2007; Javaheri & Eslami, 2002) including some post-

buckling (Shen & Li, 2008; Ke et al., 2009) and vibration analysis (Yang et al., 2006; Park & Kim, 2006).

Since these have been mostly on the stability of FGM plates, post-buckling work is yet to be seen on the onset

of interactive buckling and localization of sandwich struts with FGM cores. These secondary instabilities have

been shown to destabilize the approximately neutrally stable post-buckling response of sandwich struts and its

effects are magnified in the presence of eccentric loads (Yiatros & Wadee, 2011), geometrical imperfections

(Wadee, 2000) or buckling-driven delaminations (Wadee & Blackmore, 2001; Wadee 2002).

Herein, the sandwich strut model from Wadee and Hunt (1998) is revisited and adapted for sandwich

struts with elastic cores made from functionally graded materials. The original model captures mode

interaction and the resulting localization in the elastic range between overall and local buckling modes in

simply supported sandwich struts with a soft, linear elastic orthotropic core material (Fig. 2(a)). The model

considers pre-buckling uniform (in y) end compression (ΔL), it uses a combination of the Rayleigh–Ritz

method for the overall buckling modes, represented by W(x) and θ(x), and continuous, initially unknown,



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functions, w(x) and u(x), for the interactive mode (Fig. 2(b)). It is based on Timoshenko beam theory (TBT)

and nonlinear stability theory (Hunt & Wadee, 1998). Other variations of this model were also developed,

including models using the Reddy-Bickford Theory (RBT) (Reddy, 1984), which is a well-known higher order

shear deformation bending theory. These were employed to examine interactive buckling in sandwich panels

for different geometric and loading configurations, attaching particular importance to the onset of the

instabilities. The RBT and TBT models were first compared for sandwich struts in Wadee et al. (2010), while

sub-models for beam-columns were presented in Yiatros & Wadee (2011), for the case of axial loads being

applied offset to the neutral axis. The present work on modelling interactive buckling in sandwich panels with

an FGM core material represents the principal novel contribution. Section 2 presents the development of the

nonlinear variational model, with numerical results and validation being presented in Section 3. Finally,

Section 4 concludes by discussing possible future model enhancements, in particular accounting for the effects

of face–core delamination.

2. Model development

2.1 Geometry and kinematics

Consider a simply supported sandwich strut of length L and breadth c comprising two identical

isotropic stiff face plates of thickness t, Young’s modulus E and Poisson's ratio ν, that are separated by a weak

orthotropic core material of thickness b that potentially has different Young’s moduli in the axial direction Ex,

the transverse direction Ey and an associated shear modulus Gc; the Poisson’s ratios in x and y directions are νx

and νy respectively. A stiff, thick end-plate is attached for load application and support purposes as shown in

Fig. 2(a). Uniform end shortening prior to buckling is represented by ΔL, while the overall (Euler) buckling

mode is modelled using discrete generalized coordinates that are associated with predetermined functions with

respect to x for the lateral displacement and section rotation as shown in Eq. (1):

, (1)

where qs and qt denote the amplitudes of sway and tilt respectively; different values for these allow for the

development of shear strains in the core (Hunt& Wadee, 1998) in Eq. (2) and represent a Timoshenko beam

(TBT) model for bending. Shear strains have been found to be vital for the interaction of the two buckling

modes:

sin , coss t

x xW q L q

L L



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(2)

Local plate buckling is only considered for one of the two face plates since it is assumed that the overall mode is

triggered first. During initial post-buckling, the subsequent stress at mid-span reaches the critical compressive

stress for local plate buckling in the most compressed face plate and the localized interactive mode is triggered,

see Fig. 2(b).

Unlike the overall mode that is described by predetermined displacement functions, the interactive mode

has initially unknown displacement components for the more compressed face plate with w(x) and u(x) being the

lateral and in-plane components respectively. The interactive mode is assumed to vary in x and linearly in y,

thus:

(3)

Finally, the relative compressive strain defining the non-trivial fundamental path is modelled by Δ.

2.2 Formulation

As indicated earlier, the core is made from an FGM where the stiffness varies from the neutral axis of

bending outwards. Currently, a quadratic stiffness distribution is assumed such that:

(4)

where X is the base material property such as the Young’s or shear moduli (Ex, Ey or Gc), while XFGM gives the

variation of the property in the transverse direction y. The above function is quadratic in y ensuring that the

variation of properties is symmetric in the transverse direction to avoid any bias in the direction of overall

buckling. The reason for selecting a quadratic gradation is that it provides the simplest nonlinear continuous and

symmetric property variation, hence keeping the complexity of the pilot model to a minimum. The two

parameters, α and β, determine the nature of the gradation. As seen in Fig. 3(a), increasing , increases the rate

of the gradation quadratically, thus reducing the step change in shear stiffness at the face plate–core interface.

Setting to zero leads to uniform properties across the section. The Poisson’s ratio of the material is assumed

to remain constant, i.e. it is assumed to be unaffected by the stiffness gradient.

2.3. Energy formulation

TBT cos .s t

xq q

L

2 2, ( ); , ( ).

2 2

b y b yw x y w x u x y u xc c

b b

2

2 ,( )FGM

yb

X y X



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The system's total potential energy comprises the strain energy of the strut integrated over its volume

minus the work done by the applied load. Assuming the plane stress condition, the potential energy of the strut

is readily integrated over the breadth of the strut. Energy contributions include bending energy from the face

plates:

(5)

where EI=Ect3/[12(1-ν2)] and primes denote differentiation with respect to x, accounting for overall bending of

both face plates and local bending at the bottom face plate only; membrane energy from the face plates is given

by the expression:

(6)

where D=Etc/2 and

(7)

denote the axial membrane strains in the top and bottom face plate respectively. Moreover, there is a significant

contribution from the core deriving from direct and shear strains:

(8)

with material parameters Cx, Cy, and G being defined thus:

(9)

where Ex is the core Young’s modulus in the x-direction, Ey the core Young’s modulus in the y-direction and Gc

the shear modulus of the core. Core strains are defined in Eqs. (10)-(12):

(10)

(11)

(12)

2 2

0

1'' ' d ,

22 '

L

b EI W w xU

2 2

0d ,

L

m xt xb xU D

21' , ' ' '

2 2 2xt xbb b

u w

2/22 2

20 /2

1d d ,

21

21

L b

c x x y y x y x y c xyb x y

U E E E Gy

y xb

, , ,

22 1 2 1yx c

xx y x y

y

E bcE bC C

cG

G bc

212

, ,' c cx

u wx

x y yx

, ,xy

cx ywy

', .c c

xy

w uW

xx y

y



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The work done is the product of the applied load multiplied by the displacement at the point of application, in

the direction of the load, thus:

(13)

where the first term Δ is purely a uniform (in y) compressive strain, defined as a generalized coordinate, which

gives the non-trivial fundamental (pre-buckling) equilibrium path. The second term comprises the contribution

purely from the overall buckling lateral displacement W if the strut was inextensional. The third term comprises

an additional in-plane displacement at the point of load application, where y=0, due to the incremental local

buckling in-plane displacement u. Since u is assumed to be linear in y, the in-plane displacement contribution

from local buckling at the centroid is u/2, see Fig. 4. Moreover, since the final end-shortening contribution is

equal to [u(0)-u(L)]/2, and hence it can be written as the final term in Eq. (13) within an integral expression.

Assembling all the energy terms and integrating over the depth, the total potential energy is given as an integral

over the length of the strut:

(14)

The difference between the current model and the benchmark model in Wadee & Hunt (1998) lies in the fact

that the material parameters (core Young’s and shear moduli) are not constant, but are functions of y. Setting all

the interactive buckling terms, w(x) and u(x), and their derivatives to zero, the critical overall buckling load can

be found through linear eigenvalue analysis:

(15)

where . As expected, when and , the critical buckling load returns to the homogeneous core

sandwich strut critical load presented in Wadee & Hunt (1998). Having determined the critical load, it is then

necessary to minimize the entire expression of V to determine the governing equilibrium equations to obtain the

2

0

1 1' ' d ,

2 2

LP P W u x

0

LPP

0

LL

42 2 2 2 2 2 4 2

20

2 42 2 2 4 2 2

2

2

2

1 1 1'' sin 2 ' ' '

2 4 2

1 1 11 1sin sin ' ' ' ' '

2 2 20 84 3 10

1 7' '

4 60

2 ' ' ' 2L

s

t t x

xEI w D w w u

L L

b x b x bq q u w w

V q u u w

u wL L L L

u w

C

2 4 22 2

2

2 2 2

22 2 2

2

1 3 1' sin ' ' sin

12 12 20 2

1 1 '' ' '

2 2

2

cos

c 2o 's

10 3

t t

s y x x y x x

s t

b xq

C C

G q

b xu q u w

L L L L

x w w wP q u kw u u

L b b

x u uqq

bw

L b2

2

1cos '

2 12

'd .

10 3

s t

x uw

L b

x

q

wG

2 22C

2 2 2

/ 3 / 2 /122,

3

40 28 240 / 2

20 / 20 3 /1260 /

c x

x c

bc C EtcGEIL Etc

PG cC

/b L 1 0



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post-buckling response. Thus is achieved through the application of the calculus of variations (Fox, 1987),

which yields two coupled, nonlinear and non-autonomous ordinary differential equations for w(x) and u(x),

respectively. Moreover, by minimizing V with respect to each generalized coordinate, an equivalent number of

equilibrium equations in the form of integral constraints are also derived. The complete system of equations is

given thus:

(16)

(17)

(18)

(19)

and

(20)

The system of equations is solved within the numerical continuation software AUTO (Doedel & Oldeman, 2007),

which is well known for its capability of pinpointing bifurcations in the solution space and trace subsequent

equilibrium paths. Since the model considers geometric configurations where the overall buckling mode is

triggered first, the solution process is initiated from the primary bifurcation PC at the overall buckling critical

load. The secondary bifurcation PS can be found by increasing qs from the critical state; when located, the

2 32

2 2

22

3

2

3 1'''' 2 '' sin ' ' ' sin '' ' '' cos

2 2 2 2 6

3 1 2 11 7 1'' ' ' sin ' '' ' '' '' ' ''

5 2 3 6 140 120 15

t t x t

t

t

b x b x b xw D w q u w w q u C w u q

L L L L L L

b xw w u q w w u w u w w

L

E

L

I

q2 2

2

' 1 '''sin ' cos 2 ''

40 12 3 10

' 2sin 0,

12 3 10

y x y x x

s t

b x x u w ww w kw C C w

L L L L b b b

x uqG q G

L L b

3

2

7 3'' ' '' 2 cos

3 10 2 60 6 20

1 '2 co

2

s ' 0,2 12

x x xt

s t y x

C C Cb xu D w w D q D

L L

G x u wq w

b Lq C

b b

2 22 2

0

3 1 3sin ' '

2 6 20 12 2 6 20

1cos ' d 02 ,

2 12

Lx x

t s tqC Cb b x

D GL u w DL L L

x

q

ux

L

q

wGb

42

0

1' 2 d 0,

2 12

L

s s s t

EI uq PL q L q q w x

L bG

2

2 2

0

212

1 '' ' '

2 12

2

2 03 0

.1

x y x

L

x y xD

L D C C PL

wu w uC C



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applied load becomes the principal parameter and the post-buckling path is evaluated; see Fig. 3(b) for a

schematic representation of the response.

3. Numerical study, validation and discussion

In the pilot study herein, a sandwich strut with L = 100 mm and b = 5.1 mm is examined. The face

plates of thickness 0.5 mm are made from aluminium alloy (E = 68.9 kN/mm2 and = 0.3). The base Young's

modulus in both x and y directions in the core (when α = 1 and β = 0) is 199 N/mm2 and the shear modulus 83

N/mm2. The Poisson's ratio for the core is taken as 0.2 in both directions regardless of the value of the two

parameters (α, β) that define the functional gradation of the core material, as shown in Eq. (4).

The analytical model is validated using numerical simulations developed with ABAQUS. The core

material is discretized with the 2-dimensional solid element CPS4R, which is a 4-noded bilinear element with

reduced integration, hourglass control and an aspect ratio close to unity. The core is discretized vertically in

horizontal strips, each one element wide similarly to the model presented in Apetre et al. (2008) (see Fig. 5(a)).

The gradation in the material properties in cross-section is approximated by having a sufficiently large number

of strips of homogeneous yet different properties. For the range of values used in this study it was found that

20 strips across the depth of the strut were sufficient to provide a reasonably smooth transition in the properties

and more importantly avoid unwanted numerical errors caused by shear locking (Belytschko et al., 2000). The

key property of each strip is the value of the property at its mid-height, which is derived from the gradation

function given in Eq. (4). The face plates are modelled as stringers bonded to the edges of the existing core

while specifying suitable engineering properties. The 2-dimensional linear Timoshenko beam element B22 is

used to discretize the stringers. Owing to the small thickness of a face plate (t = 0.5 mm), in conjunction with

the large length and the need to maintain a reasonable element aspect ratio, this approach is less computationally

demanding, since only one beam element through the thickness of the face plate is used without compromising

accuracy (Zenkert, 1995; Léotoing et al., 2002), especially in the cases of thin face plates and has therefore been

implemented for all the tested struts.

To capture the overall buckling mode and the corresponding critical load, a linear perturbation analysis

was performed; Fig. 6(a) shows the increase of the critical load from the linear perturbation analysis, which

correlates very well with the analytical model. Qualitatively, regarding the mode shape, for the FE simulation

where >0, which implies a higher stiffness further away from the neutral axis, the shear strain is seen to be

maximum with a localized peak at the neutral axis and approximately constant further from it, see Fig. 5(b). The



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first buckling mode was then used as an initial perturbation (imperfection) in the model geometry in the

subsequent nonlinear analysis, by employing the modified Riks method (ABAQUS, 2006). An imperfection

amplitude of L/10000 was selected since it is sufficient to trigger the instability, yet small enough to give results

close to the perfect case. It should be noted that although no local imperfection is explicitly incorporated in the

model, a secondary instability and buckle pattern localization is subsequently observed from a self-generated

local imperfection that evolves during the post-critical (Riks) analysis.

When the parameter α is set to unity, while β is increased from zero (i.e. a homogeneous core) to 12,

the critical loads predicted by the model increase with an increasing value of β. This is expected since β

increases the stiffness of the core, thereby contributing to the buckling resistance of the strut. For the position of

the secondary bifurcation which triggers interactive buckling, the gap between the critical and secondary

bifurcations decreases with increasing β, as seen in Fig. 6(b). Although for smaller values of β this compares

very well with the finite element model (Fig. 7), for larger values of β the comparison becomes divergent. This

is attributed to the fact that the analytical model does not fully account for the shear stiffness energy near the

neutral axis of bending as a result of the “plane sections remain plane” assumption of TBT. This leads to an

underestimation of the total potential energy of the strut which is reflected by the two bifurcations coming closer

together as the critical load increases, a feature that is not replicated in the finite element model. However,

beyond the secondary bifurcation the post-buckling path becomes unstable owing to the presence of the

localized mode being confined only to one face plate; this is, however, observed clearly in both the analytical

and finite element models with the unloading shown to match reasonably well, as shown in Fig. 7.

4. Concluding remarks

An analytical model based on Timoshenko beam theory, has been adapted to investigate interactive buckling in

sandwich struts with functionally graded cores. The merits of a functionally graded core can be realized by

assigning a certain material property where it is most needed, without creating steep changes. This gradual

change in the properties across the depth of the strut can minimize the risk of delamination in composite

components, such as sandwich structures, where there is a large interlaminar shear stress between the two

laminates in the cases where the required adhesive is not so strong. If the stiffness of the softer laminate

gradually increases towards the interface, the danger of shear discontinuities appearing is significantly reduced.

On the other hand, the reduction of the stiffness at places where it is not necessarily required could be utilized in

an optimization strategy in systems with weight constraints. The model has been tested with struts where the



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shear stiffness in the core is graded quadratically and symmetrically about the neutral axis, increasing at the face

plate–core interface. Increasing the degree of gradation results in an increase in the overall buckling load, but

shows a reduction in the gap between the critical and secondary bifurcations due to the underestimation of the

shear stiffness since a fundamental assumption of TBT is that plane sections remain plane.

The analytical model has been validated using a finite element model implemented in ABAQUS that

exhibited good correlation in terms of the position of secondary bifurcation points and the subsequent post-

buckling paths. The reason for this can be attributed to the in-plane displacement field being approximately

linear except very locally near the neutral axis. Further work would be necessary to generalize the models for

different gradation functions and more loading cases, but this study has provided a useful baseline for

determining which models best suit the simplest gradations. This class of model is envisaged to be able to

validate an enhanced model of a previously developed approach (Abali et al., 2012) that has been implemented

with the aid of the open source finite element software FEniCS (Logg et al., 2011) based on continuum

mechanics formulations. The major advantage of this numerical approach is that the gradient properties can be

readily implemented as a function over the thickness of the core rather than modelling the variation with several

layers, thus leading to a more accurate representation whilst having greater numerical efficiency. Furthermore,

by using Green–Lagrange strains, being conjugated to the second Piola–Kirchhoff stresses, when deriving the

weak formulation of equilibrium, geometric nonlinearities are already intrinsically part of the formulation.

Hence, the post-buckling paths can be traced without any stepwise iteration or need for initial imperfections.

Additionally, by solving the problem using continuum mechanics principles interlaminar stresses at the interface

between the core material and the face plates can be determined, subsequently interfacial damage may be

predicted and its progression may be traced. This approach would provide an excellent way of assessing the

advantages of using FGMs and allow tailoring to optimize the performance of such sandwich structures.

Figure 1



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Fig. 1: Interactive buckling between the overall (Euler-type) and local (plate) buckling on a sandwich strut

100mm long and 5.1mm deep.

Figure 2a Figure2b

(a)

(b)

Fig.2: (a) The simply supported sandwich strut in elevation and cross-section. (b) Modal descriptions. From the

top: uniform (pre-buckling) end shortening due to pure compression represented by ΔL, sway mode represented

the overall buckling deflection W(x), tilt mode represented by θ(x) and the local modes, w(x) and u(x) becoming

non-trivial beyond the secondary bifurcation.

Figure 3a Figure3b

(a)

(b)

Fig. 3 (a) Variation in material properties X through the core depth. (b) A graphical representation of the

equilibrium response: (1) fundamental path; (2) overall buckling; (3) interactive buckling. The critical load is

PC and the quantity is the end shortening at the critical load. Similarly and qsS refer to the values of end

shortening and the sway amplitude at the secondary bifurcation, respectively.

Figure 4

Fig. 4 The total end shortening contributions for work done. Top: pre-buckling end shortening, Middle: end-

shortening from lateral displacement W from overall buckling, considering the strut as inextensional and

Bottom: in-plane displacement u(x,0) from the interactive mode.

Figure 5a Figure5b

CC SS



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(a)

(b)

Fig. 5: (a) The variation ( ) in the core shear modulus modelled for FE simulations in ABAQUS. The

same applies for core Young’s modulus in both directions. The analytical model is represented by the

continuous function and the numerical model with a step gradation. (b) A schematic of the in-plane

displacement due to overall buckling as this was seen in the finite element simulations somewhere along the

length of the strut.

Figure 6a Figure 6b

(a)

(b)

Fig. 6: (a) Comparison of the critical loads from the analytical model and FE simulation for different values of β

from 0 to 12. The analytical model results are represented by “○” and the numerical model results by “∆”. (b)

The equilibrium paths from the analytical model: load versus end shortening per unit length, for a strut 5.1 mm

deep, α =1 and different values of β from 0 to 12.

Figure 7a Figure7b

(a)

(b)

Fig. 7: The comparison of the equilibrium paths for the analytical model and the FE simulation for struts with

different degrees of stiffness of gradation (a) = 3, (b) = 6.

References

1; 4



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Modeling of Interactive Buckling in Sandwich Struts with Functionally Graded Cores

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