Accurate buckling load calculations of a thick orthotropic sandwich panel

Composites Science and Technology 72 (2012) 1134–1139

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Accurate buckling load calculations of a thick orthotropic sandwich panel Wooseok Ji, Anthony M. Waas ⇑ Department of Aerospace Engineering, Composite Structures Laboratory, University of Michigan, Ann Arbor, MI 48109, United States

a r t i c l e

i n f o

Article history: Received 9 November 2011 Received in revised form 15 February 2012 Accepted 23 February 2012 Available online 15 March 2012 Keywords: A. Sandwich B. Mechanical properties C. Computational mechanics C. Anisotropy C. Sandwich structures

a b s t r a c t This paper is concerned with the buckling of thick sandwich panels with orthotropic elastic face sheets bonded to a linear elastic orthotropic core. When such panels are analyzed for axial load carrying capacity, it is now commonplace to adopt the finite element method to carry out computations. The accuracy of the numerical results will depend not only on roundoff and algorithmic errors, but additionally on the approximations made in computing the incremental (second order) work associated in computing the change of configuration from the unbuckled to the buckled state. Here we show that, particularly for orthotropic thick sandwich structures, large errors can be incurred in computing buckling loads with available commercial software, unless the proper work conjugate measures of stress and strain with their stress-dependent tangential moduli are used in the buckling formulation. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction It is well known that various types of stress measures have a work-conjugate relation with a specific type of deformation measure. Bazant [1] pioneered the development of a unified formulation that uses proper sets of a work-conjugate pairs of stress and corresponding strains, with the corresponding tangential moduli that should be used in calculating buckling loads of solid bodies. When a specific type of stress measure is chosen to describe a deformation state of a solid, the corresponding strain measure and tangential moduli can be found exactly through [1],

ðmÞ ¼ ij

i m=2 1 h F ki F kj  dij m

ð1Þ

where F ij is the deformation gradient, dij the Kronecker’s delta, and ðmÞ is the parameter indicating a specific formulation set, e.g., m ¼ 2, for Green-Lagrangian strain tensor, m ¼ 1, for Biot’s strain tensor, and m ¼ 2, for the Almansi strain tensor. Each strain measure is paired with its work-conjugate stress expression. For example, the Green-Lagrangian strain tensor and the second Piola–Kirchhoff stress tensor are paired through a work-conjugate definition. The corresponding expression for the correct tangential modului for such work-conjugate pairs are available in [1], and it reads,

  1 ðmÞ ð2Þ C ijkl ¼ C ijkl þ ð2  mÞ rik djl þ rjk dil þ ril djk þ rjl dik 4

ð2Þ

When a continuum deformation state is described through a certain set of stress, strain and corresponding constitutive ⇑ Corresponding author. Tel.: +1 734 764 8227; fax: +1 734 763 0578. E-mail address: [email protected] (A.M. Waas). 0266-3538/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compscitech.2012.02.020

relations, the work-conjugacy requirement should be maintained throughout the solution phase to guarantee the same end result regardless of the the different stress and strain measures chosen for the problem description [1]. Of course, these measures can change between different phases of a solution process, but the work conjugacy requirement cannot be violated at any stage. This fact is especially essential in numerical implementation of these formulations, for example, in finite element analysis (FEA) of solids. The problem that is of concern in this paper is the computation of the buckling loads of thick, axially compressed, sandwich composite panels. These panels are currently being used in a broad range of weight-critical applications in the aerospace, marine and infrastructure industries. Since finite element based structural analysis is now the mainstay in structural design calculations in industry, it is imperative to have formulations and implementations that produce accurate predictions, especially for thick, multi-material structural designs, such as sandwich panels. In the following section, the finite element (FE) formulation for an bifurcation buckling problem is derived first in a general three-dimensional setting. The FE formulation is applied to predict buckling loads of a thick sandwich panel with orthotropic elastic constituents. Results from the ABAQUS commercial software are compared against analytical predictions to examine the significance of the proper formulation of the buckling problem. 2. Finite element formulation of the bifurcation buckling problem The general three-dimensional formulation of the buckling problem of concern, with the details of the associated finite element (FE) formulation is presented in [2,3]. The FE formulation is

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introduced here in a refined fashion to clearly point out the significance of the correct formulation for a buckling problem of a thick sandwich, orthotropic composite panel. All three constituents of the sandwich panel (top face sheet, core and bottom face sheet) are treated as orthotropic media. The general form of the equilibrium equations of a solid in the buckled state may be obtained from the principle of virtual work, as

Z VB

Sij

@dui dV B  @X j

Z

ti dui dCB 

CB

Z VB

bi dui dV B ¼ 0

ð3Þ

^ 0ij

Sij ¼ r þ r

0

ð5Þ



Z

@dui dV B  @X j

CB

VB

CB

The constitutive model corresponding to r

ð2Þ ij

according to [4] is

rijð2Þ ¼ C ijkl ekl

ð10Þ

where eij ¼ ð1=2Þðui;j þ uj;i Þ is the linearized small strain in terms of the displacement fields. Additionally, using the symmetry properties of stress, rij ¼ rji , we can rewrite the foregoing equation as, Z Z Z Z deij C ijkl ekl dV B þ r0kj ui;k dui;j dV B  t0i dui dCB  b0i dui dV B ¼ 0 VB

VB

CB

VB

ð11Þ

u ¼ Nq

t 0i dui d

Z CB

t 0i dui dCB 

Z

B

C 

VB

0 bi dui dV B

Z VB

0

bi dui dV B þ

Z

¼0

V

r^ 0ij B

VB

VB

@dui dV B @X j

Z ð6Þ

ð12Þ

with N being the assumed shape functions and q being the vector of nodal displacements, each term of Eq. (11) is transformed into,

Z

Substitution of Eqs. (4) and (5) into Eq. (3) yields,

V

VB

ð9Þ

Z

bi ¼ bi þ bi

r0ij B

ð2Þ

where the stress tensor, rij , is work-conjugate to the GreenLagrangian strain tensor. Substitution of Eq. (8) into Eq. (7) yields Z Z Z Z 0 B B B 0 0 rð2Þ du dV þ r u du dV  t du d C  bi dui dV B ¼ 0 i;j i;j i kj i;k i ij

When the displacement fields are discretized as

t i ¼ t 0i þ t 0i

Z

ð8Þ

ð4Þ

^ 0ij is the incremental stress tensor that will be discussed in where r detail later. Suppose that the surface traction, and the body force in Eq. (3) are also decomposed into initial (prebuckled) and perturbed (buckled) quantities, such that

0

0 r^ 0ij ¼ rð2Þ ij þ rkj ui;k

VB

where Sij is the nominal stress, dui the displacement field, t i the nominal traction on the boundary CB of the initial state, bi the body force per unit volume of the base state, and V B is the volume of the body in its reference (base) configuration. The general deformed configuration of a solid, which includes a reference stress state, a pre-buckled but loaded state and a buckled state, is schematically illustrated in Fig. 1 with quantities associated with each deformation state indicated in the figure. The nominal stress in the buckled state, Sij , can be decomposed into the Cauchy stress in the prebuck^ 0 by the relation [4] led state and the incremental stress r 0 ij

moduli in Eq. (2). It was shown in [5,6] that if the incremental elastic moduli in a column are kept constant (i.e., independent of stress), the Green-Lagrangian strain and its associated formulation ^ ij is rewritten as [4] must be used. In this case, r

VB

deij C ijkl ekl dV B ¼ dqT

Z VB

dui;j r0kj ui;k dV B ¼ dqT

dui t 0i dCB ¼ dqT

Z

 BT CBdV B q ¼ KM

"Z

#  T @N @N r0 dV B q ¼ KG @x @x VB

NT t0 dV B

ð14Þ

 ð15Þ

VB

Z

ð13Þ



Since the prebuckled state is an equilibrium state, we obtain the incremental virtual work equation that governs the buckling problem, and it reads

Z

Z

Here, B is the derivative of N with respect to x. Simplifying Eq. (11) with the discretized terms, the FE variational formulation is obtained as

V

r^ 0ij B

@dui dV B  @X j

Z CB

t 0i dui dCB 

Z VB

0

bi dui dV B ¼ 0

ð7Þ

^ 0ij , can be defined in various The incremental stress tensor, r ðmÞ ways by using rij that is paired with its work-conjugate strain, Eq. (1) [1,4]. In this presentation, the set that corresponds to m ¼ 2, is employed, and this corresponds to constant tangential

VB

0

dui bi dCB ¼ dqT

0

NT b dV B

VB

dq½ðKM þ KG Þq  F ¼ 0

ð16Þ

ð17Þ

where KM is the material stiffness matrix, KG the geometric stiffness matrix, and F is the force vector. The condition corresponding to bifurcation buckling of the structure is obtained when the second variation of the energy vanishes, resulting in,

ðKM þ KG Þdq ¼ 0

ð18Þ

Thus, the FE formulation for the eigenvalue buckling problem reduces to

det ½KM þ kKG 

ð19Þ

where k is the buckling load factor. 3. Results and discussion

Fig. 1. Deformation of a solid from its base configuration to the buckled state. Quantities associated with each state are also shown.

Fig. 2 illustrates the loading and boundary configuration of the sandwich panel to be studied here. Throughout the presentation, the superscript ‘f’ represents quantities associated with the face sheets, and ‘c’ with the core. We consider symmetric, orthotropic thick sandwich composite panels, gaining wide use in large ship structures, launch vehicle structures, and bridge decks. For

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xij ¼

  1 @ui @uj  2 @xj @xi

ð21Þ

Therefore, Eq. (7) now becomes,

Z V

rijð0Þ dui;j dV B þ B 

Z

Z VB

ðr0kj xik  r0ik ekj Þdui;j dV B 

Z CB

t 0i dui dCB

0

VB

bi dui dV B ¼ 0

ð22Þ

and can be further simplified to

Z Fig. 2. Configuration of a sandwich panel (plane strain) under external pressure loading.

V

rijð0Þ deij dV B þ B 

purposes of calculation, two cases are used for numerical computation. In Case 1, the face sheets are isotropic and bonded to an orthotropic core, and in Case 2, the face sheets and core are both orthotropic. The face sheets correspond to typical thick unidirectional carbon fiber reinforced polymer matrix laminates. The top and bottom face sheets are assumed to have the same geometric and material properties. The sandwich panel is subjected to a plane strain deformation in the xy plane. The FE model is created using a quadrilateral plane strain elements as shown in Fig. 3a. Sufficiently fine meshes are used in order to have acceptable accuracy in computing buckling loads. Fig. 3b shows a typical deformed configuration of the sandwich panel. The FE formulation described in the previous section is utilized to compute the buckling load of the sandwich structure. The Green-Lagrange strain and its work-conjugate incremental stress with the corresponding constant elastic modulus forms the basic set used in the FE formulation to solve the bifurcation buckling problem. To demonstrate the violation of work conjugacy in the buckling formulation (not paying attention to work conjugate pairing of stress and corresponding strain measures), the incremental stress that is paired with the logarithmic strain m ¼ 0 is used. Eq. (8) is now written as

r^ 0ij ¼ rijð0Þ þ r0kj xik  r0ik ekj

ð20Þ

where

Z

CB

Z V

t0i dui dCB 

r0kj ui;k dui;j dV B  B

Z

VB

Z VB

2deij r0jk eik dV B

0

bi dui dV B ¼ 0

ð23Þ

Note that Eq. (23) is identical to Eq. (11) when the correct conð0Þ ð0Þ stitutive model, rij ¼ C ijkl ekl , is used. Bazant [1] showed that a consistent formulation of the bifurcation buckling problem must be independent of the choice of the finite strain tensor. To demonstrate the error caused by the improper buckling formulation, the constitutive model with the constant elastic modulus, rð0Þ ij ¼ C ijkl ekl , is used in Eq. (23). In the following discussion, the formulation that corresponds to Eq. (11) is denoted as ‘Formulation A’ and the formulation that results in Eq. (23) with the constant modulus is referred to as ‘Formulation B’. Buckling loads from the two formulations are compared to examine the importance of the correct buckling formulation. When the constant modulus is used in Eq. (23), the term assoR ciated with the volume integral 2 V B deij r0kj eik dV B does not vanish. This extra term results in a stiffer KG in Eq. (19), when the composite is under compressive loading (r0kj < 0), leading to higher buckling load predictions as shown in [2]. Fig. 7 of [2] shows that over-prediction of buckling loads is significant as the core becomes stiffer since the effect of rij in the extra term is not negligible for stiffer cores. 3.1. Case 1: orthotropic core with isotropic face sheets In this paper, to investigate the effect of the core properties on the buckling load prediction, sandwich panels with an orthotropic core are considered. Fig. 4 compares buckling loads when the aspect ratio of the sandwich panel corresponding to Case 1, L=h is

2.4 Formulation A

2.2

Formulation B ABAQUS

2

Pcr /PB

(a)

1.8

1.6

1.4

(b) Fig. 3. (a) Finite element modeling of a sandwich panel; (b) typical deformed configuration of the sandwich structure.

0

20

40

60

80 c

100

120

140

c

E xx /E yy Fig. 4. Case 1: Orthotropic core with isotropic face sheets. Buckling load comparc ison as a function of the core stiffness ratio, EEcxx . yy

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W. Ji, A.M. Waas / Composites Science and Technology 72 (2012) 1134–1139

4

set to 3, and the stiffness ratio of the orthotropic core is varied. Buckling loads in Fig. 4 are normalized by P B defined as

Formulation A

PE

ð24Þ

1 þ PE =GA

Formulation B

3

where PE ¼ p2 EI=L2 is the Euler’s buckling load, EI the equivalent bending stiffness of a sandwich panel, and GA is the equivalent shear resistance of a sandwich panel [6]. The properties of the constituents are given in Table 1, where the face sheets are made of an isotropic material, [7]. In Fig. 4, the buckling loads from Formulation B (Table 2) as well as ABAQUS differ considerably from the results of Formulation A as the ratio between the longitudinal and transverse moduli of the core is varied. This indicates that the effect of the non-vanishing term, pointed out earlier, becomes significant when the core undergoes shear, extensional and transverse deformations. It is noted that many ‘‘higher order’’ sandwich panel theories do not include the complete deformation state of the core and this omission may or may not be accurate depending on the stiffness mismatch between the core and face sheet properties. Reinforced cores such as K-core and X-core in the aerospace sector have significant core properties, which are not to negligible when compared against the face sheet properties. These foam cores are reinforced through the insertion of carbon-fiber-reinforced plastic (CFRP) pins oriented in a three dimensional truss network. K-cores have the tips of pins pressed flat onto the core surface for flexibility in design and manufacturing process in laminating face sheets. In X-cores, the pin tips are evenly penetrated into face sheets to achieve a superior bonding between face sheets and the core [5]. Fig. 5 shows the buckling load comparison when the aspect ratio between the panel length to width varies with the material properties given in Table 3. Again, formulation B and ABAQUS shows large errors in predicting the buckling load as the sandwich panel become thicker. Buckling loads of the thick sandwich panels are greatly influenced by the core deformation. When the core is made of an orthotropic material, it appears that errors in predicted buckling loads increase since the effect of core orthotropy is more significant for a thick sandwich configuration. Fig. 5 also shows the results from ABAQUS (Table 4), whose bifurcation buckling model is formulated based on the Jaumann’s stress rate. It should be noted that the results from ABAQUS and the results from the incor-

Table 1 Material properties of the face sheets and the core in Fig. 4. Case 1

Property

Value

Face sheet

Ef (GPa) h (mm)

30 0.3 0.1

Ecyy (GPa)

4  Gcxy

Gcxy c xy f

m

Gf =1000 0.25

h (mm)

0.8

mf f

Core

(GPa)

Pcr /PB

PB ¼

ABAQUS

2

1

0

0

2

4

6

8

10

12

L/h Fig. 5. Case 1: Orthotropic core with isotropic face sheets. Buckling load comparison as a function of the panel aspect ratio, hL.

Table 3 Material properties of the face sheets and the core for the sandwich panel in Fig. 5. Case 1

Property

Value

Face sheet

Ef (GPa)

30 0.3 0.1

mf f

h (mm) Ecxx Ecyy Gcxy c xy c

Core

(GPa)

400  Gcxy

(GPa)

4  Gcxy

(GPa)

m

Gf =1000 0.25

h (mm)

0.8

Table 4 Tabular data of the predicted buckling loads shown in Fig. 5. L/h

Formulation A

Formulation B

ABAQUS

2 4 6 8 10

2.1716 1.3666 1.1563 1.0655 1.0094

3.1387 1.7781 1.4820 1.3449 1.2535

3.1028 1.7700 1.4738 1.3449 1.2535

rect formulation B, agree with each other, as shown in the examples.

3.2. Case 2: orthotropic core with orthotropic face sheets

Table 2 Tabular data of the predicted buckling loads shown in Fig. 4. Ecxx =Ecyy

Formulation A

Formulation B

ABAQUS

20 40 60 80 100 120

1.5275 1.5599 1.5873 1.6121 1.6351 1.6568

1.6399 1.7786 1.9078 2.0296 2.1450 2.2546

1.6310 1.7786 1.9078 2.0198 2.1349 2.2443

Fig. 6 compares buckling loads as a function of the stiffness ratio of the core between the longitudinal and transverse direction for a sandwich panel with orthotropic face sheets and an orthotropic core. The material properties used for the results in Fig. 6 are listed in Table 5. The orthotropic material properties of the face sheets are obtained from [8]. The aspect ratio of the sandwich structure, L=h, is set to 3 for this example. In Fig. 6, the buckling loads from Formulation B (Table 6) and ABAQUS are higher than the results from Formulation A as the core stiffness ratio increases. It appears that with the high orthotropy of the core, the error from the erroneous formulation (formulation B) becomes significant. Furthermore, the incorrect formulation predicts much higher buckling

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W. Ji, A.M. Waas / Composites Science and Technology 72 (2012) 1134–1139

4

2.6

Formulation A

Formulation A

2.4

3

ABAQUS

Pcr /PB

Pcr /PB

Formulation B

Formulation B

2.2

2

1

2

1.8

ABAQUS

0

20

40

60

80 c

100

120

0

140

0

2

4

6

c

8

10

12

L/h

E xx /E yy Fig. 6. Case 2: Orthotropic core and orthotropic face sheets. Buckling load c comparison as a function of the core stiffness ratio, EEcxx .

Fig. 7. Case 2: Orthotropic core and orthotropic face sheets. Buckling load comparison as a function of the panel aspect ratio, hL.

yy

Table 7 Material properties of the face sheets and the core for the sandwich panel in Fig. 7.

Table 5 Material properties of the face sheets and the core in Fig. 6.

Case 2

Property

Face sheet

Efxx (GPa)

107

107

Efyy (GPa)

15

Gfxy (GPa)

4.3

Case 2

Property

Value

Face sheet

Efxx (GPa)

Core

Value

Efyy (GPa)

15

Gfxy (GPa)

4.3

mfxy

0.3

h (mm)

f

0.1

Ecxx (GPa)

480  Gcxy

Ecyy Gcxy c xy c

(GPa)

4  Gcxy

Ecyy (GPa)

4  Gcxy

(GPa)

m

0.3

h (mm)

0.1

f xy f

Core

Gfxy =200

Gcxy (GPa)

Gfxy =200

m

0.25

mcxy

0.25

h (mm)

0.8

hc (mm)

0.8

Table 8 Tabular data of the predicted buckling loads shown in Fig. 7.

Table 6 Tabular data of the predicted buckling loads shown in Fig. 6. Ecxx =Ecyy

Formulation A

Formulation B

ABAQUS

20 40 60 80 100 120

1.8877 1.9160 1.9397 1.9610 1.9807 1.9993

1.9644 2.0654 2.1611 2.2531 2.3423 2.4289

1.9644 2.0654 2.1506 2.2426 2.3316 2.4289

loads, leading to a highly non-conservative situation from a design stand-point. This can have disastrous and dangerous implications. Fig. 7 shows the buckling load comparison with a variance of the aspect ratio between the panel length to thickness. Table 7 lists the material properties used in Fig. 7. Formulation B and ABAQUS overpredict the buckling loads as the sandwich structure becomes thicker (Table 8). As discussed earlier, the effect of the non-vanishing term, resulting from the incorrect formulation of the bifurcation buckling problem, becomes significant when the core deformation is not negligible. Ji and Waas [2] showed that when the core has almost the same order of stiffness as the face sheets, predicted buckling loads from improper formulations result in large errors. One cannot simply neglect the axial load carrying

L/h

Formulation A

Formulation B

ABAQUS

2 4 6 8 10

2.9281 1.5800 1.2609 1.1361 1.0687

3.8147 1.9119 1.5155 1.3564 1.2668

3.7839 1.9119 1.5155 1.3674 1.2668

capacity of the core. When the orthotropic phase is introduced in the sandwich structures, the incorrect formulation is again seen to greatly overpredict buckling loads even for sandwich panels with relatively soft cores as shown in the examples here. 3.3. Effect of othotropy in predicting buckling loads Indeed, material orthotropy greatly influences the elastic stability of a solid as studied in [3]. It is shown that, in buckling of compressed highly orthotropic columns that are very soft in shear, the use of an improper formulation (constant modulus coupled with the Jaumman’s stress rate) can cause large errors, as high as 100% of the critical load, even if the strains are small [3]. Figs. 2 and 3 of [3] show that buckling loads of an orthotropic strip computed from Formulation B and ABAQUS significantly differ from the

W. Ji, A.M. Waas / Composites Science and Technology 72 (2012) 1134–1139

100

4. Conclusion

Error (%)

80

60

40

20

0

1139

0

200

400

600

800

1000

E xx/E yy Fig. 8. Error in predicting buckling load of an orthotropic strip as a function of the xx strip stiffness ratio, EEyy .

Table 9 Material properties of a orthotropic strip in Fig. 8. Property

Value

Eyy (GPa) Gxy (GPa)

2  Gxy 7.17 0.29 1 3

mxy h (mm) L/h

results from Formulation A when the aspect ratio of the strip changes. In this paper, the effect of material orthotropy (characterized using the stiffness ratio between the longitudinal and transverse direction) on the buckling load predictions have been characterized. The material properties of the orthotropic strip in Fig. 8 are summarized in Table 9. The aspect ratio of the strip is set to 3. The error in Fig. 8 is defined as the relative error of the buckling loads between Formulation A and Formulation B. As shown in Fig. 8, the error increases as the material orthotropy is enhanced.

In this study, the correct bifurcation buckling formulation for a thick orthotropic sandwich structure in a general 3D setting, has been presented. By examining the buckling behavior of sandwich panels, with isotropic and orthotropic face sheets bonded to an othotropic core, under plane strain deformation, it has been shown that the finite element formulation leads to accurate predictions of buckling loads when the correct conjugate stress–strain pair with its associated moduli are properly used. It is also shown that improper formulations can produce results that are in error, and that these errors grow as the structure becomes thicker and material orthotropy becomes significant. Both of these effects are important for design calculations of modern sandwich panels used in the marine, aerospace and infrastructure industries. The error incurred by using the commercial FE package, ABAQUS, is also demonstrated and it is recommended that FE methods for studying stability problems in thick composites, sandwich structures, and other orthotropic materials must be cognizant of the approximations associated with what is meant by ‘thin’ and ‘thick’, since such approximations require a consideration of both material properties and geometry. In particular, along with the geometric slenderness ratio, the magnitudes of the ratios of axial modulus to shear modulus, and axial modulus to transverse modulus must be considered. Furthermore, the recommendations made in the earlier work [3], with respect to commercial codes is once again emphasized. References [1] Bazˇant ZP. A correlation study of formulations of incremental deformation and stability of continuous bodies. Trans ASME, Ser E, J Appl Mech 1971;38(4): 919–28. [2] Wooseok Ji, Anthony Waas. 2D elastic analysis of the sandwich panel buckling problem: benchmark solutions and accurate finite element formulations. Z Angew Math Phys 2009;61(5):897–917. [3] Wooseok Ji, Anthony Waas, Zdenek Bazant. Errors caused by non-workconjugate stress and strain measures and necessary corrections in finite element programs. J Appl Mech – Trans ASME 2010;77(4). [4] Baz˘ant Zdene˘k P, Cedolin Luigi. Stability of structures: elastic, inelastic, fracture and damage theories. New York: Oxford University Press; 1991. [5] Baz˘ant ZP, Beghini A. Which formulation allows using a constant shear modulus for small-strain buckling of soft-core sandwich structures. J Appl Mech 2005;72(5):785–7. [6] Bazˇant ZP, Beghini A. Stability and finite strain of homogenized structures soft in shear: sandwich or fiber composites, and layered bodies. Int J Solids Struct 2006;43(6):1571–93. [7] Fleck NA, Sridhar I. End compression of sandwich columns. Compos – Part A: Appl Sci Manuf 2002;33(3):353–9. [8] Fagerberg L. Wrinkling and compression failure transition in sandwich panels. J Sandw Struct Mater 2004;6(2):129–44.