Multiaxial failure surface of PVC foams and monitoring of deformation bands by three-dimensional digital image correlation

Multiaxial failure surface of PVC foams and monitoring of deformation bands by three-dimensional digital image correlation

Accepted Manuscript

Multiaxial failure surface of PVC foams and monitoring of deformation bands by three-dimensional digital image correlation Francesca Concas, Stefan Diebels, Anne Jung PII: DOI: Reference:

S0022-5096(18)30874-3 https://doi.org/10.1016/j.jmps.2019.06.008 MPS 3644

To appear in:

Journal of the Mechanics and Physics of Solids

Received date: Revised date: Accepted date:

8 October 2018 14 June 2019 15 June 2019

Please cite this article as: Francesca Concas, Stefan Diebels, Anne Jung, Multiaxial failure surface of PVC foams and monitoring of deformation bands by three-dimensional digital image correlation, Journal of the Mechanics and Physics of Solids (2019), doi: https://doi.org/10.1016/j.jmps.2019.06.008

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Highlights • Mechanical properties of PVC foams were obtained for several loading

CR IP T

cases. • 3D-DIC by an 8-camera system allowed the inspection of deformation bands during tests

• Deformation bands in torsion tests are strictly related to the orthotropy of the foam.

AN US

• Combined axial-torsional tests revealed the asymmetry of the failure sur-

AC

CE

PT

ED

M

face.

1

ACCEPTED MANUSCRIPT

3D-DIC by an 8-camera system

strain fields and deformation bands

simple tests

superposition of torsion

(c)

(d)

(e)

(a)

(a)

(b)

(d)

successive torsional loading

(e)

AC

CE

PT

ED

M

AN US

compressive preloading

(c)

failure surface

diagrams and failure limits

(b)

CR IP T

compressive preloading

multiaxial tests

2

ACCEPTED MANUSCRIPT

Multiaxial failure surface of PVC foams and monitoring of deformation bands by three-dimensional digital image correlation

a Saarland b University

CR IP T

Francesca Concasa,b , Stefan Diebelsa , Anne Junga,∗

University, Applied Mechanics Campus A4.2, Saarbr¨ ucken 66123, Germany of Cagliari, Department of Mechanical, Chemical and Materials Engineering, via Marengo 3, Cagliari 09123, Italy

AN US

Abstract

Closed-cell polyvinylchlorid foams are among the most used core materials for sandwich composites. Therefore, a deep knowledge of their multiaxial behaviour is fundamental. More precisely, a literature review proves the lack of investigations concerning combined tensile-torsional loadings. In the present work simple

M

tests, namely tension, compression and torsion have been performed for determining failure loads. Afterwards, multiaxial tests have been executed by applying different percentages of the previously calculated failure loads as preload

ED

and subsequent superposition of torsion, keeping simultaneously the axial load constant. For each multiaxial test, failure data have been collected in order to construct the failure surface for the foam material. The whole specimen surface

PT

has been inspected by means of the three-dimensional digital image correlation, in order to detect the failure of the specimen, as well as the onset and the devel-

CE

opment of deformation bands. Monitoring of the specimen has been performed for each main loading case. Afterwards, the analysis of deformation bands has been coupled with stress-strain diagrams and linked with the arising of buckling

AC

or fracture and the instrinsic orthotropy of the foam. Keywords: foam material (B), mechanical testing (C), three-dimensional digital image correlation, buckling (A), deformation bands ∗ Corresponding

author Email address: [email protected] (Anne Jung)

Preprint submitted to Journal of the Mechanics and Physics of Solids

June 17, 2019

ACCEPTED MANUSCRIPT

1. Introduction

CR IP T

Closed-cell polyvinylchlorid (PVC) foams are among the most used core materials for sandwich composites due to their impact withstanding associated with light weight [1] and lower cost, if compared with other polymeric foams [2]. 5

Due to manufacturing processes PVC foams exhibit an orthotropic behaviour.

Therefore, it must be distinguished between through-thickness [3, 4] and in-

plane [3, 4, 5] mechanical properties. PVC foams are used in the ship building

AN US

as well as in the aerospace industry [6, 7, 8, 9]. A further typical application as

sandwich core consists in wind turbine blades [10]. In this particular case, PVC 10

foam cores undergo multiaxial loading conditions as indicated in the literature [4, 11]. Therefore, the investigation of failure surfaces for polymeric foams is of crucial importance, especially in the case of combined shear and axial loads, namely compression and tension.

15

M

The construction of yield or failure surfaces has been extensively addressed in the scientific literature: a throughout summary of contributions of authors fo-

ED

cusing on polymeric foams has been reported by Shafiq et al. [12], which further enhanced the overview in this field by adding more than one hundred of multiaxial tests on PVC foams, several of which are triaxial tests both in tension and

20

PT

compression. In their literature review, Shafiq et al. [12] disclosed also that less attention has been paid to combined compression-shear and especially tensile-

CE

shear tests. Among the works listed in Shafiq et al. [12], the contribution of Gdoutos et al. [5] shall be mentioned, since they dealt with the construction of failure surfaces along the through-thickness direction for two different den-

AC

sities of a PVC foam core grade. Because of the limited thickness of available

25

panels, they constructed tubular specimens for combined tension-torsion tests by gluing pieces of panels onto each other followed by machining. Ring specimens were used for combined compression-torsion tests. Another important contribution on the construction of failure surfaces for polymeric foams is the work of Christensen et al. [13], which performed combined tension-torsional 4

ACCEPTED MANUSCRIPT

30

and compression-torsional loading on tubular specimens having the axis parallel to the through-thickness direction of PVC foam panels and the height lower than the thickness panel. Therefore, usage of glue for the specimen construction

CR IP T

was avoided. In both of the latter works, tests were performed by an universal testing machine, analogously to Altenbach and Kolupaev [14] and Fahlbusch et 35

al. [15], which executed combined axial-torsional tests on polymethacrylimide

(PMI) foams. Multiaxial i.e. compression-shear tests on PMI foams were performed also by Benderly and Putter [16] using a flexural test fixture. In Hoo Fatt

et al. [17] a method for performing triaxial tests by using an Arcan fixture, a

40

AN US

pressure vessel and a universal testing machine is presented. Nonetheless, only

pure compression, pure shear and mixed compression-shear cyclic tests were conducted on PVC foams. Donnard et al. [18] achieved combined compressionshear tests by a hexapod fixture on an expanded polypropylene foam. Most of the previously mentioned authors developed an analytical constitutive model of the failure surface and compared it with the experimental collection of data referring to multiaxial tests. One of the most recent works in this field was

M

45

presented by Ayyagari et al. [19], which is compared by Shafiq et al. [12] with

ED

gathered experimental data.

Multiaxial testing procedures and the construction of yield surfaces were tackled for metal foams as well. One of the most exhaustive collection of yield data was recently presented by Jung and Diebels [20, 21] for open-cell aluminium foams

PT

50

characterised by three different pore sizes. Specimens having a square-section

CE

and a height to width ratio of two were loaded in compression, tension, torsion and combined tension-torsion and compression-torsion with five different percentages of the axial yield load. Yield data were finally analysed according to the yield function developed by Bier et al. [22, 23, 24]. Wu et al. [25] and Zhang

AC

55

et al. [26] compared results of simulated multiaxial cases on the proposed threedimensional Voronoi models with experimental tests such as simple compression [26] and further uniaxial and biaxial tension [25]. Simulations of triaxial tests were performed as well. Luo et al. [27] presented numerical yield surfaces given

60

by simulations of biaxial and triaxial compression on a finite element model 5

ACCEPTED MANUSCRIPT

R taken from micro-tomographic images of an Alporas foam.

Recent contributions concerning mechanical properties of PVC foams were given by Rajput et al. [28] and Miyase and Wang [29]. Both works deal with the inves-

65

CR IP T

tigation of the compressive modulus focusing on specimen shape and size ratios. In Rajput et al. [28] different methods for strain measurement were applied

and resulting values were compared. In Miyase and Wang [29] the tensile and shear modulus along the through-thickness and in-plane direction were studied,

too. Experimental analyses of the behaviour of PVC foams undergoing impact

loading were recently issued [30], taking also into account different densities of the foam core for sandwich panels [31].

AN US

70

The present contribution focuses on the construction of the failure surface for R PVC foams known with the trade name of Divinycell HP100. At the begin-

ning simple tests i.e. tension, compression and torsion, are performed. Failure thresholds are set, based on the type of the imposed simple load. Afterwards, 75

combined compression-torsion and tension-torsion tests are conducted taking

M

into account different percentages of the previously specified failure load, in order to collect data for the construction of the failure surface. Simple tests and

ED

combined tension-torsion and compression-torsion are monitored by an 8-camera system for the execution of 3D-DIC on the whole surface of the cylinder and 80

the observation of trends of deformation bands. Gathered failure data for PVC

PT

foams are adapted by the yield criterion proposed by Bier et al. [22, 23, 24].

CE

2. Materials and methods 2.1. PVC foams and specimen preparation

AC

R The Divinycell HP100 foam panels were supplied by Diab AB (Laholm,

85

Sweden). Closed-cell PVC foams classified as HP afford good performances at high temperatures. Therefore, they are suitable for sandwiches manufactured by the prepregs method [32]. According to a communication from the manufacturer, HP is the acronym of ”High Performance” referring to the abovementioned feature. The number in the nomenclature specifies the nominal den-

6

ACCEPTED MANUSCRIPT

90

sity i.e. 100 kg/m3 . Among technical data [32] from the manufacturer, main mechanical properties are provided as well. Compared to the all-purpose [33] H PVC foam type, whose microstructure was broadly examined by many works

CR IP T

for different values of the nominal density, the microstructure of the HP PVC foam was investigated only by Colloca et al. [34] and Luong et al. [35]. Nev95

ertheless, the noticeable elongation of cells towards the direction of the panel’s

thickness, mentioned also by Zhang et al. [3], was not considered. Therefore, a statistical investigation of the cell size taking into account the two main directions is advantageous. A small piece of the PVC foam was cut from a 40 mm

100

AN US

thick panel and its main surfaces, namely the ones perpendicular and parallel to the through-thickness direction, were inspected by optical microscopy (digital

microscope VHX-500F of Keyence Ltd., Osaka, Japan). Three characteristic measures for the cell size are considered (see Fig. 1 (a)), a and b refer to the longest and the shortest size, respectively of the elliptic shape of the cells observed on the surface parallel to the through-thickness direction. c concerns the size of the isotropic cell structure observed on the surface perpendicular to

M

105

the through-thickness direction. Fig. 1 shows further micrographs of surfaces

ED

oriented parallel to the two main directions. For each characteristic size 20 measurements are done for statistics. Measurements are performed by placing points on the cells contours which appear as most distinguishable in each image acquired by the microscope, as shown in Fig 1(b-c). Based on the set magnifi-

PT

110

cation, the corresponding dimension is given by the software of the microscope.

CE

The measurement results are summarised in Table 1. The values of b and c are consistent with values obtained by Colloca et al. [34] and Luong et al. [35] in terms of averages and standard deviations. The value of a is less than twice the values of both b and c, as remarked also by Zhang et al. [3] for a different

AC

115

R Divinycell foam core grade. As is widely known, the elongation of the cells is

the reason for the general anisotropic behaviour of foams [5, 36]. Further examinations of the foam surface parallel to the through-thickness direction prove that cells do not feature any elongation approaching the outer surface of the

120

purchased foam panel, which was mentioned also by Zhang et al. [3]. Micro7

ACCEPTED MANUSCRIPT

graphs from Fig. 1 (b)-(c) and relating magnifications show that undamaged cell walls are highly reflecting and broken walls tend to bend, which made it difficult to recognise cells distinctively. This phenomenon is more evident in the surface

AC

CE

PT

ED

M

AN US

CR IP T

parallel to the in-plane direction.

8

ACCEPTED MANUSCRIPT

CR IP T

(a)

(b)

ED

M

AN US

c

a b

AC

CE

PT

(c)

Figure 1: Schematic view of a panel with magnification of the different cell morphologies along the two main directions (a), images obtained by the digital microscope of a foam piece cut by a band saw in parallel with the in-plane (IP ) (b) and the through-thickness (TT ) (c) direction, with dimensions relating to the characteristic sizes. The arrows indicate some cells

that are partially covered by bending walls.

9

characteristic

measure

size

µm

a

568.07 ± 144.65

b

344.43 ± 106.67

c

334.96 ± 83.30

CR IP T

ACCEPTED MANUSCRIPT

Table 1: Average values and standard deviations of the characteristic sizes measured on 20 cells by optical microscopy.

The cylindrical specimens shown in Fig. 2 (a) were used in order to prevent

125

AN US

warping during torsional tests. The specimens featured a larger diameter to both ends for ensuring a proper clamping in the universal testing machine by means of the steel bushings depicted in Fig. 2 (b) specifically fabricated for such

a specimen geometry. In accordance to the work of Jung and Diebels [20] the 130

gauge part of the cylindrical specimen had a height to diameter ratio of two, which allowed the exemplification of shear strain calculations. The specimen

M

size and especially the maximal diameter of the specimen part assigned to the clamping had been chosen taking into account the thickness of the available

in-plane direction.

AC

CE

PT

135

ED

panels, i.e. 40 mm. The axis of the cylindrical specimen is parallel to the

10

ACCEPTED MANUSCRIPT

(b)

AN US

CR IP T

(a)

Figure 2: Specimen geometry (a) and cutaway of bushings mounted on the specimen during

M

testing (b).

Cylindrical specimens were made by loading pieces of foam into a lathe and

ED

using a grinding wheel as tool. Since jaws of the chuck would irretrievably damage the foam surface, different steel bushings were used, whose internal profiles were specifically made for inserting the foam workpiece during each stage of the specimen manufacturing. So the foam could be conveniently clamped

PT

140

into the lathe chuck. Main steps of the manufacturing procedures are portrayed

CE

schematically in Fig. 3. First, pieces featuring a square section were cut by a band saw from a 40 mm thick PVC foam panel (see Fig. 3 (a)). Subsequently, the foam piece was loaded into the chuck of the lathe by means of a squaresection bushing. The tail-stock of the lathe was also needed for fixing properly

AC

145

the workpiece. At this stage only half of the piece was roughly shaped by the grinding wheel in order to achieve a round section with a diameter of 38 mm (Fig. 3 (b)). Afterwards, the second half of the workpiece was shaped as well into the round section by using a further bushing for clamping the already machined

150

part. The tail-stock was temporarily removed (Fig. 3 (c)). Last, the final shape 11

ACCEPTED MANUSCRIPT

of the specimen was obtained by using two further bushings and placing the tail-stock as depicted in Fig. 3 (d). Finally, the whole surface of the specimen was painted by a speckle pattern for

155

CR IP T

performing the three dimensional digital image correlation (3D-DIC). A white varnish and subsequently a black varnish by an aerosol spray are applied as background and random speckle pattern, respectively. In order to fulfil the requirements for an efficient and reliable speckle, a thick coating of varnish

had to be applied as background on the specimen. If the coating is not thick enough, the black paint for speckling would spread across the foam cells and a proper contrast would not be obtained. Preliminary tests demonstrated that

AN US

160

the mechanical properties of painted specimens does not differ from those of unpainted specimens.

(b)

ED

M

(a)

(d)

CE

PT

(c)

AC

Figure 3: Schematic sequence referring to the different specimen manufacturing phases.

2.2. Experimental probing of the failure surface

165

All tests were performed using the ElectropulsTM E10000 universal testing

machine of Instron Ltd. Pfungstadt, Germany, characterised by an actuator stroke of 60 mm, a load cell capacity of 10 kN for axial loading and 100 Nm for torsional loading. Loading cycles and triggering for the registration of images 12

ACCEPTED MANUSCRIPT

were defined by means of the Instron WaveMatrix software. Simple tensile and compression tests were executed by the actuator driven in displacement con170

trol with a speed of 0.1 mm/s, which correspond to a strain rate of 0.002 1/s.

CR IP T

Torsional tests were performed in rotational control with an angular velocity of 0.5 deg/s, with a shear strain rate of 0.002 1/s. For both combined tensiletorsion and compression-torsion tests, the actuator was driven in load control

with a speed of 5 N/s up to reach a specific percentage of the prescribed axial 175

failure load, i.e. 25%, 50% and 75%, subsequently torsion was applied in rota-

tional control with the above mentioned angular velocity while the axial load is

AN US

simultaneously kept constant. Possible visco-elastic effects are neglected. Preliminary tests had shown a longer extension of the failure curve branch towards

the positive axis of the first principal stress invariant, consequently, a combined 180

tension-torsion loading case reaching the 90% of the tensile failure load was further conducted. Moreover, preliminary tests had exhibited a low scattering of

guarantee for statistics.

M

the failure data. Therefore, three tests for each loading case were performed to

Two three-jaw chucks were mounted on the universal testing machine in order to clamp the specimens having a round section. As stated previously, two bush-

ED

185

ings were inserted between the jaws and the specimen. The usage of bushings is essential, since a common method such as the application of adhesive on the

PT

specimen (e.g. [4, 5]), could influence the material behaviour during the tests and was not sufficient for a reliable clamping. In other cases, scientific literature does not even mention methods applied for clamping the specimen. The work

CE

190

of Sirikuk et al. [9] should be remarked as they used a special fixture for performing tensile and torsional tests on PVC foams. The normative [37] proposes

AC

the usage of peculiar grips.

13

ACCEPTED MANUSCRIPT

3D-DIC by 8 cameras: experimental set up

upper bushing

three-jaw chucks

CR IP T

cross marks

AN US

specimen

camera (8)

M

lower bushing

Figure 4: Experimental set-up. -3-

195

ED

The 3D-DIC strain analysis was achieved on the whole cylindrical surface of the specimen by usage of an 8-camera system consisting of Manta G-235B cam-

PT

eras (Allied Vision Technologies GmbH, Stadtroda, Germany). The cameras were mounted on aluminium section bars as represented in Fig. 4. Furthermore, the cameras were arranged in such a way as to compose a regular octagon, on

CE

whose vertices one camera was located. The image acquisition and the subse-

200

quent local strain evaluation were performed by the software Istra 4D v. 4.4.2 of Dantec Dynamics GmbH (Skovlunde, Denmark). Eight images were acquired

AC

simultaneously by triggering. All images featured a resolution of 1936 × 1216 pixels. Once the camera system was properly set around the specimen, a calibration was needed in order to evaluate the projection parameters [38]. This

205

procedure was done by rotating a glass plate with a printed pattern in the field of view of each camera pair. Afterwards, intrinsic and extrinsic parameters were

14

ACCEPTED MANUSCRIPT

calculated. The lighting of the specimen plays a crucial role for a correct evaluation of the local strain since localised reflections must be avoided, which might easily occur for such a specimen geometry. Three lamps had been included in the experimental set-up. Light sources had been located in order to illuminate

CR IP T

210

the whole cylindrical surface of the specimen as uniformly as possible. Despite the fact that the usage of multi-camera systems is challenging regarding the preparation of the equipment and cameras, it gives a wider insight on the de-

formation mechanism of the specimen notably for compression tests, in which 215

the occurrence of the characteristic curvature for a specimen subjected to buck-

AN US

ling is random. Therefore, the usage of only one pair of cameras might not be sufficient for a proper monitoring of deformation bands. Every 90◦ , four crossshaped marks were depicted in addition to the speckle pattern at the central

section of the specimen, as can be seen in the magnification of the specimen 220

from Fig. 4. Each mark was located towards the centre of a single camera pair and its size was comparable to the subset size defined in the software. The

M

marks worked as immediately recognisable starting point for the evaluation of local strains in each camera pair.

225

ED

The software Istra 4D conducted the compound of the cylindrical surface and the evaluation of local displacements and local strains from the acquired images. Afterwards, the obtained local strain fields were imported into the software

PT

c (Thierry Zamofing, Paul Scherrer Institute, Switzerland) in orh5pyViewer

der to unwrap the above-mentioned cylindrical surface. In Fig. 5, the geometry

CE

obtained by Istra 4D and the subsequent unwrapping are shown for a tensile 230

test. Only for tensile tests, it is advisable to include the conical part of the specimen in the evaluation of the local strains, as depicted in Fig. 5. Evident

AC

black areas on the unwrapped specimen in Fig. 5 (b) are due to unavoidable reflections and glares, which arose on the contour of the specimen and hindered the monitoring of the specimen through 360◦ .

235

R Furthermore, self-written Python scripts were developed and tailored for each

loading case for dealing with:

15

ACCEPTED MANUSCRIPT

• the parsing of raw output data from the universal testing machine, • the calculation of stresses, global strains, elastic and failure parameters

CR IP T

• the fitting of the yield function taken from Bier et al. [23, 24] • the plotting of graphs

240

(b)

M

AN US

(a)

unwrapping (b).

ED

Figure 5: Undeformed specimen geometry compounded by the 8-camera system (a) and its

PT

For uniaxial loading average stresses σ are calculated by

σ=

F A

(1)

AC

CE

and global strains are yielded by the Green-Lagrange formulation [39] 1 ε= 2

"

L L0

2

#

−1

(2)

F indicates the force applied to the specimen, A is the cross-section in the

gauge part of the specimen. L and L0 denote to the current height and the

245

initial height of the cylindrical gauge part, respectively. For each loading case, nominal stresses are calculated since they are related to the initial cross-section

16

ACCEPTED MANUSCRIPT

of the specimen. For torsional loading the maximal shear stress τ and the shear global strain γ occurring in the contour of the circular section are given by r j

and

(3)

CR IP T

τ =M

r γ = πθ , (4) L respectively. M is the applied torque, r is the radius of the cylindrical gauge

250

part, j is the polar moment of inertia and θ indicates the shear angle. The

AN US

above-mentioned stress and global strain values are directly calculated from

data registered by the Instron WaveMatrix software during the tests. The elastic moduli, i.e. Young’s modulus, compression modulus and shear modulus, 255

are calculated from the tangent of the initial elastic part of the stress-global strain curve.

M

√ The failure surface was built in the invariants plane J2 − I1 , where I1 is the √ first principal invariant of the stress tensor and J2 is the square root of the second deviatoric invariant. For uniaxial tension, compression and torsion as well as for combined tension-torsion and compression-torsion, the hydrostatic

ED

260

PT

stress σ0 , the first principal invariant I1 and the square root of the second √ deviatoric invariant J2 are simplified as follows

AC

CE

σ0 =

p J2 =

σ11 + σ22 + σ33 σ11 → σ0 = , 3 3 I1 = 3σ0 → I1 = σ11 ,

r h i 1 2 2 2 2 + τ2 + τ2 (σ11 − σ0 ) + (σ22 − σ0 ) + (σ33 − σ0 ) + τ12 23 31 2 r 2 p σ11 2 + τ12 → J2 = 3

(5)

(6)

(7)

where σ11 and τ12 are given by Eqs. (1) and (3), respectively. In order to construct the failure surface, stress limits for each loading case had to be 17

ACCEPTED MANUSCRIPT

265

established. For compressive tests, the peak yield stress was assumed as failure limit. For tensile and torsional tests, due to the visco-elastic behaviour, the stress for which fracture occurred in the specimen, was considered as failure

CR IP T

limit.

3. Results and discussion 270

3.1. Compression and tensile tests

Stress-global strain diagrams for tensile tests and for compression tests per-

AN US

R HP100 are shown in Fig. 6, formed each on three specimens of the Divinycell

where compressive stresses are displayed as positive in order to be compared directly with tensile stresses. In the following figures, compressive stresses are 275

reported as positive for the sake of simplicity. Compression tests were stopped when the descending actuator reached a distance of 3 mm from the lower limit switch, which corresponds to making the specimen enter an advanced stage of

M

the plateau. The curves exhibit a relatively low reproducibility in terms of both the pseudo-elastic [40, 41] initial part and peak yield stresses. Table 2 shows that standard deviations for the afore seen parameters reach about 10% of the mean values.

ED

280

The tensile tests shown in Fig. 6 exhibit a higher reproducibility, i.e. standard

PT

deviations for the failure load and the failure stress are about 1% of the mean value. The Young’s modulus shows a slightly higher uncertainty with a stan285

dard deviation equal to 3%. As is evident from Fig. 6, the specimens undergoing

CE

a tensile loading reach higher stresses compared to the specimens subjected to compression in the elastic and the initial plateau stage. Furthermore, the peak

AC

yield stress observed in compression is absent in tensile loading. Table 2 summarises the averages and the standard deviations of failure loads, failure stresses

290

and elastic moduli for both loading conditions, compression and tension. The average failure load for the tensile tests is 230% of the average failure load for the compression tests. The same value is thereby obtained for the ratio between average failure stresses. The elastic moduli for both, compression and tension,

18

ACCEPTED MANUSCRIPT

display analogous values of the average and the standard deviation. This is 295

in contrast to the assumption of bimodularity normally taken in the literature [42], which described different elastic moduli for tension and compression load-

CR IP T

ings. The obtained mechanical properties consider the in-plane direction, hence a comparison with the mechanical properties provided by the manufacturer [32]

M

AN US

cannot be made, since the latter concerns the through-thickness direction.

ED

Figure 6: Stress-global strain curves for three specimens subjected to compression loading and tensile loading, respectively. Failure stresses are marked by small ticks.

Tension

Failure load FF [N ]

624.41 ± 61.53

1433.42 ± 14.82

Failure stress σF [M P a]

1.27 ± 0.13

2.92 ± 0.03

52.82 ± 5.62

53.24 ± 1.61

CE

PT

Compression

Elastic modulus [M P a]

Table 2: Averages and standard deviations of the failure load, the failure stress and the

AC

elastic modulus for specimens subjected to compression and tensile loading, respectively.

300

Fig. 7 shows the specimen subjected to compressive loading prior the test

and during different stages of the test. The five chosen stages refer to different points of the stress-global strain curve, i.e. a point taken from the elastic phase,

19

ACCEPTED MANUSCRIPT

the peak yield point and three points of the plateau phase. The buckling occurs once the specimen entered into the plateau phase. Based on the varnish, the 305

specimen might lose fragments of the varnish coating during severe buckling.

CR IP T

For compression tests there was no fracture and the peak yield stress is considered as failure limit. This assumption is related to the applicative use of PVC

foams as core of sandwich composites. The plateau stress of foams is needed for calculating their absorption energy potential [34, 43] in applications where an 310

impact loading is involved. According to Barsotti et al. [43], the plateau stress

is considered as compressive strength for crushable foams. In the cylindrical

AN US

specimens, because of the buckling occurrence during the plateau stage, the stress is not constant, whereas the peak yield stress is unequivocally determined from the diagram of Fig. 7. The usage of a different geometry exclusively for 315

compression testing of PVC foams would give back a more constant value of the plateau stress. In that case, the plateau stress could be considered as failure limit. Since its value would be lower than the peak yield stress, the resulting

M

failure criterion would be more restrictive. However, using the same cylindrical geometry for all tests allows to perform a direct comparison between diagrams for the different loading cases. Thus, a more effective investigation on the me-

ED

320

chanical properties can be accomplished. Fig. 8 shows local Lagrangian vertical strain fields on the unwrapped cylindrical

PT

geometry for the five global strain stages, which are highlighted in the stressglobal strain diagram of Fig. 8 (f). The beginning of the buckling can be noticed in the unwrapped surface of Fig. 8 (c), in which high local strain gradients ap-

CE

325

pear and the main deformation band in the upper part of the local strain map corresponds to the local strain localisation in the specimen shown in Fig. 7 (c).

AC

The unwrapped surface of Fig. 8 (c) concerns the first point in the plateau phase. As the applied load increases, i.e. as the buckling grows, regions with a positive

330

value of the local vertical Lagrangian strain appear. Furthermore, deformation bands merge with each other up to form distinct areas, characterised by a positive or negative value of the local Lagrangian vertical strain. This is due to the buckling, i.e. zones of the specimen near the convex and the concave part 20

ACCEPTED MANUSCRIPT

(b)

prior to testing

(c)

(d)

(e)

(b)

(c)

(d)

(e)

AN US

(a)

CR IP T

(a)

- -

M

strain localisation

Figure 7: Images acquired by one camera during different stages of the compression test. The arrow indicates the beginning of the local strain localisation. The failure stress is marked by

ED

a cross in the compressive stress-compressive global strain diagram.

of the curve undergo tensile and compressive stresses, respectively. For severe buckling (Fig. 8 (e)), the correlation is not accomplished in some localised areas

PT

335

of the specimen surface, that involves both greater distortions of the specimen

CE

and the already-mentioned lose of varnish coating pieces. Deformation bands prior to the buckling are perpendicular to the axis and the generatrix of the cylindrical specimen. After the entrance of buckling, deformation bands appear mostly curvilinear in the unwrapped specimen (see Fig. 8 (c)-

AC

340

(e)). Hence, deformation bands become perpendicular only to the generatrix of the specimen.

21

(b)

(c)

(d)

CR IP T

(a)

M

AN US

local Lagrangian vertical strain

ACCEPTED MANUSCRIPT

(f)

ED

(e)

(b) (c)

(d)

CE

PT

(a)

-2Figure 8: Local Lagrangian vertical strain maps (a)-(e) for five average global strain values

AC

taken from the compression stress-global strain curve (f).

Fig. 9 (a) depicts images of the specimen prior to tensile testing and after

the failure. The tensile failure appears clearly brittle and perpendicular to

345

the loading axis. This was also found by Christensen et al. [13] for tubular specimens of a different PVC foam core grade.

22

(e)

ACCEPTED MANUSCRIPT

Figs. 9 (b)-(f) show local strain fields for five stages on the unwrapped cylindrical surface of the specimen. For all performed tensile tests, the fracture developed in a restricted area of the specimen, which contains one end of the cylindrical part and is very near the conical part of the specimen. Hence, the conical

CR IP T

350

surface is partially included in the region of interests. Deformation bands are

perpendicular to the cylinder axis and are equal for each camera pair. This

means that misalignment of the bushings did not occur during the test. Based on preliminary tests, a slight misalignment of the bushings is not uncommon 355

immediately prior to the tensile failure. The misalignment of the bushings would

AN US

lead to an abnormal elongation of the tensile stress-global strain curve.

The highest local strain gradient is found in the lower deformation band, where the fracture takes place. The lower deformation band is also perpendicular to the specimen axis, whereas deformation bands appear slightly wavy in the upper-central part of the specimen.

AC

CE

PT

ED

M

360

23

ACCEPTED MANUSCRIPT

prior to testing

(a)

(c)

(d)

after failure

(e)

(b) (a)

2

AN US

(d)

(e)

M

-4-

(c)

4

ED

local Lagrangian vertical strain

(b)

3

CR IP T

1

AC

CE

PT

(f)

Figure 9: The tensile stress-global strain curve and images of the specimen prior to testing and after failure (a) with reference to the local Lagrangian vertical strain maps (b)-(f) for five global strain values. The failure stress is marked by a cross in the stress-global strain diagram.

24

5

ACCEPTED MANUSCRIPT

3.2. Torsion tests Shear stress-shear global strain curves are depicted in Fig. 10 for three specR imens of the Divinycell HP100 foam subjected to pure torsional loading. The

365

CR IP T

specimens have shown a similar mechanical behaviour to the tensile loading,

characterised by the deviation of the curve from the elastic proportionality between stresses and global strains (without peak yield stress) and brittle fracture.

Since the relative standard deviations of the mechanical properties summarised

in Table 3 are lower than 6%, torsion tests are more reproducible than compression tests. The already mentioned misalignment of the bushings did not take

AN US

place in any test, which is presented in this work.

PT

ED

M

370

Figure 10: Shear stress-global strain curves for three specimens subjected to torsion loading.

AC

CE

Failure stresses are marked by small ticks.

Torsion Failure torque MF [N m]

5.44 ± 0.32

Failure shear stress τF [M P a]

1.77 ± 0.10

Shear modulus G [M P a]

28.97 ± 1.28

Table 3: Averages and standard deviations of the failure torque, the failure shear stress and the shear modulus for three specimens subjected to torsional loading.

25

ACCEPTED MANUSCRIPT

Fig. 11 (a) shows images of the specimen prior to testing and after failure. Analogously to compression and tensile loading cases, in Fig. 11 (b)-(f) results of the 3D-DIC for a torsion test are shown. Local Lagrangian tangential shear

375

CR IP T

strain fields are represented in the unwrapped cylindrical geometry for five different values of the shear global strain highlighted in Fig. 11 (a). The local strain

field of the image immediately preceding the failure cannot be shown because

the correlation was not achieved any more after the fifth stage in Fig. 11 (a), i.e. Fig. 11 (f). The correlation loss is due to high distortions occurring in the

torsion test and the exit of subsets from the shared field of view of each cameras pair. The shear global strain γ = 0.43, for which the failure of the specimen

AN US

380

occurs, corresponds to a relative rotation of the bushings of 99.66◦ . Higher local strain gradients are clearly visible in two diametrically opposed wedges of the specimen. This is due to the different elongation of the cells in the foam panel, from which the specimen was manufactured. 385

The failure band has an angle of 45◦ to the loading direction outlining shear

M

failure. Since the incipient failure cannot apparently be found in deformation bands of Fig. 11, it indicates that the failure is dominated by tensile principal

ED

stresses. The above-mentioned software for the 3D-DIC evaluation Istra 4D allows to depict local Lagrangian tensile principal strain maps on the specimen. 390

The relating results are shown in Fig. 12. Deformation bands in local Lagrangian

PT

tensile principal strain maps are more pronounced than deformation bands in local Lagrangian tangential shear strain maps. Furthermore, the positioning of

CE

deformation bands in two diametrically opposed wedges of the specimen is kept. By observing the strain maps in Fig. 12 and neglecting their boundaries, where local strain data are less reliable, the failure originates in the area of the specimen with the highest local principal strain gradient. Subsequently, the fracture

AC

395

propagates as a helix.

26

ACCEPTED MANUSCRIPT

prior to testing

(a)

(c)

(d)

(e)

after failure

(f)

AN US

(d)

(e)

ED

M

-1-

(c)

(f)

AC

CE

PT

local Lagrangian tangential shear strain

(b)

CR IP T

(b)

Figure 11: The shear stress-shear global strain curve and images of the specimen prior to torsional testing and after failure (a) with reference to the local Lagrangian tangential shear strain maps (b)-(f) for five global shear strain values. The failure stress is marked by a cross in the shear stress-shear global strain diagram.

27

(b)

(c)

(d)

CR IP T

(a)

M

AN US

local Lagrangian tensile principal strain

ACCEPTED MANUSCRIPT

(f) (a)

(b)

(c)

(d)

CE

PT

ED

(e)

Figure 12: Local Lagrangian tensile principal strain -2maps (a)-(e) for five average shear global

AC

strain values taken from the shear stress-shear global strain curve (f) of the torsion test.

3.3. Combined compression-torsion and tensile-torsion tests

400

Fig. 13 shows shear stress-shear global strain diagrams for combined tests.

The procedure for combined compression-torsion and tension-torsion testing has been described in Section 2.2 by using a certain percentage of the uniaxial

28

(e)

ACCEPTED MANUSCRIPT

failure stress values determined in Section 3.1 as axial loading before the torque is continuously increased. The plateau stress of combined compression-torsion tests is in general greatly scattered (see Fig. 13 (a)). Furthermore, for a low percentage of the compression preloading, i.e. 25%, a significant scatter of

CR IP T

405

the peak yield stress can be observed. Increasing the compressive preloading percentage leads to a gradually lower peak yield stress and a slightly decreasing scatter of the data.

This latter feature is observed also for combined tensile-torsion tests in terms 410

of the final failure stress. With higher percentages of the failure tensile load,

AN US

tests exhibit a better reproducibility, i.e. a lower scatter. Furthermore, shear stress-shear global strain curves relating to percentages of 25% and 50% of the failure tensile load appear similar with each other. (a)

PT

ED

M

(b)

Figure 13: Shear stress-shear global strain curves for specimens subjected to different percentages of compression loading (a) and tensile loading (b) as preload and subsequent torsion,

CE

respectively. Failure stresses are marked by small ticks.

Local Lagrangian strain fields are obtained by the 3D-DIC for the com-

bined compression-torsion test. The preload increases as a ramp up to reach

AC

415

the 50% of the determined peak yield stress. The application of the preload is an artefact of the machine for reaching the required value of the normal stress. Afterwards, the torque is applied simultaneously to the normal load, up to reach either the limit switch of the machine or the failure of the specimen for com-

420

bined compression-torsion tests and combined tension-torsion test, respectively. 29

ACCEPTED MANUSCRIPT

Hence, the application of the normal load and the torque happens simultaneously. Strain maps and diagrams concerning the application of the preload are shown exclusively for the sake of thoroughness of information.

425

CR IP T

In Fig. 14 images of the tests with the matching global strain value in stressstrain curves are shown. The buckling does not arise for this type of multiaxial test. Because of the similarity between the shear stress-shear global strain curve and the stress-global strain curves of Fig. 8 (f), local Lagrangian strain maps are

referred to the following global strain values: the first is placed during the compressive preloading, the second is set in the elastic stage during superposition of the torsional loading, the third is set in the peak yield stress during super-

AN US

430

position of the torsional loading, the fourth and the fifth are placed in the peak yield stress during superposition of the torsional loading. A further similarity between the compression test and the combined compression-torsion test is the

AC

CE

PT

ED

M

absence of fracture.

30

ACCEPTED MANUSCRIPT

superposition of torsion

(b)

(c)

(a)

(b)

(c)

(e)

(d)

(e)

AN US

(a)

(d)

CR IP T

compressive preloading

successive torsional loading

M

compressive preloading

ED

Figure 14: Images acquired by one camera during different stages of the combined compressiontorsion test, reaching the 50% of the peak yield stress. The failure stress is marked by a cross in the shear stress-shear global strain diagram.

PT

The local Lagrangian vertical strain map for an average global strain value

435

during the compressive preloading is shown in Fig. 15. Most deformation bands

CE

arise perpendicular to the axis of the specimen. Strain fields concerning the subsequent superposition of the torsional loading are depicted in Fig. 16. The gradual disappearance of subsets is again due to their exit from the shared field of view of each cameras pair. Despite keeping the compression load constant

AC

440

during the superposition of torsion, the position of deformation bands is similar to torsion testing, with higher local strain gradients in the upper-central part of the specimen, i.e. near the rotating actuator. Deformation bands are located in two diametrically opposed wedges of the specimen.

31

local Lagrangian vertical strain

ACCEPTED MANUSCRIPT

CR IP T

(a)

AN US

(b)

M

(a)

- Figure 15: Local Lagrangian vertical strain map (a) for a global strain value taken from the

AC

CE

PT

ED

compressive preloading stress-strain curve (b).

32

(b)

(c)

(d)

CR IP T

(a)

ED

M

AN US

local Lagrangian tangential shear strain

ACCEPTED MANUSCRIPT

(e)

(b)

(c)

(d)

CE

PT

(a)

-14 Figure 16: Local Lagrangian tangential shear strain maps (a)-(d) for four shear global strain

AC

values taken from the shear stress-shear strain curve (e), which concerns the superimposed torsional load. The failure stress is marked by a cross in the shear stress-shear global strain diagram.

445

Lastly, local Lagrangian strain fields were obtained for a combined tensile-

torsion loading, in which the tensile preload reaches the 50% of the failure stress. For this loading case, the considered failure load draws the specimen to 33

ACCEPTED MANUSCRIPT

fracture. Images of the specimen during the tests are shown in Fig. 17. The first two images concerns the tensile preloading, whereas the last threes refers to the superposition of torsion by keeping the tensile loading constant. tensile preloading

superposition of torsion

CR IP T

450

(d)

(b) (c)

(b)

(c)

(d)

(e)

ED

M

(a)

AN US

(a)

(e)

tensile preloading

successive torsional loading

PT

Figure 17: Images acquired by one camera during different stages of the combined tensiontorsion test. The failure stress is marked by a cross in the shear stress-shear global strain

CE

diagram.

Fig. 18 shows local Lagrangian vertical strain fields for two global strain

values during the tensile preloading. The deformation bands in Fig. 18 (b) are

AC

typical for a simple tension test. A continuous main deformation band can be seen in the lower part of the unwrapped cylinder. Deformation bands in the

455

upper-central part are mainly perpendicular to the specimen axis. By increasing the loading, they would take the slight wavy shape as seen for the uniaxial loading in Fig. 9. Local tangential shear strain maps are shown in Fig. 19 with reference to the 34

ACCEPTED MANUSCRIPT

superposition of torsion for the last three average shear global strain values. 460

The higher local strain gradient is found in the lower part of the local strain map, which immediately precedes the failure of the specimen (see Fig. 19 (c)).

CR IP T

The corresponding deformation band indicates the forthcoming fracture. The specimen breaks as in the case of tensile fracture but the fracture of such a

combined tensile-torsion loading is propagated as a helix. The fracture towards a helix is coherent with the torsion failure.

local Lagrangian vertical strain

465

(b)

ED

M

AN US

(a)

(c) (b)

CE

PT

(a)

-10 Figure 18: Local Lagrangian vertical strain maps (a)-(b) for two global strain values taken

AC

from the tensile preloading stress-strain curve (c).

35

(b)

(c)

(d)

CR IP T

(a)

(b)

AN US

local Lagrangian tangential shear strain

ACCEPTED MANUSCRIPT

M

(a)

9 Figure 19: Local Lagrangian tangential shear strain maps (a)-(c) for three average shear

ED

global strain value taken from the shear stress-shear global strain curve (e), which concerns the superimposed torsional load. The failure stress is marked by a cross in the shear stressshear global strain diagram.

PT

3.4. Construction of the failure surface Failure stresses, which were obtained from experimental tests on HP100

CE

foams, are used for calculating invariants by means of Eqs. (6) and (7). Invari-

AC

470

ants are finally gathered in order to build the failure surface in the invariants √ plane J2 − I1 (see Fig. 20).

36

(c)

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 20: Failure data for the HP100 foam depicted on the

√

J2 − I1 plane and fitted by the

M

yield criterion proposed by Bier et al. [22, 23, 24] with extrapolation of the failure surface towards the axis of negative first principal invariants I1 .

ED

Square root of the second deviatoric invariant values relating to combined compression-torsion tests exhibit a more notable scattering in case of lower percentages of the compressive contribution, i.e. 25% and 50% in comparison to

475

PT

failure data concerning other loading cases. Whereas the square root of the second deviatoric invariants relating to uniaxial tests display generally a lower

CE

scattering and, in the particular case of pure tensile loading, an overlapping between failure data is seen. Failure invariants are subsequently fitted by means of the yield function pro-

AC

posed by Bier et al. [23, 24] in the implicit form taken from Bier and Hartmann

480

[22], 

p  F I1 , J2 = ck ln

√

eg1 (I1 ,

J2 )/(ck)

√

+ eg2 (I1 , 2

J2 )/(ck)

!

.

(8)

√
√
Both functions g1 I1 , 2 and g2 I1 , 2 are defined by Bier and Hartmann 37

ACCEPTED MANUSCRIPT

[22] as follows,

 p  p g2 I1 , J2 = J2 − A1 + A2 eA3 I1 ,

with

(9)

CR IP T

 p  q 2 g1 I1 , J2 = J2 + α (I1 − 3ξ) − k ,

(10)

A1 = k , 1−

√

1−

r2

k
I0 /((3ξ−I0 )(1+r)) ,

AN US

A2 =

(11)

A3 = ln (k/A2 ) /I0 , q 2 k = α (I0 − 3ξ) .

Thus, five independent parameters are identified, i.e. α, c, I0 , ξ, r, which are 485

R script from the experimentally evaluated by means of a self-written Python

ED

M

determined failure data. The obtained parameters are finally listed in Table 4.

α [−]

c [M P a]

I0 [M P a]

ξ [M P a]

r [−]

1.065

-22.094

3.773

0.605

-0.547

PT

Table 4: Parameters of the yield criterion presented by Bier et al. [23, 24].

CE

The yield criterion introduced by Bier et al. [23, 24] has been applied to aluminium foams for the first time by Jung and Diebels [20, 21], which demon-

490

strated that aluminium foams underwent yielding, i.e. a plastic collapse stress,

AC

for all loading cases and proved the suitability of the above-mentioned yield criterion for foams, whose yield surface is asymmetrical between the tension and compression parts. Fig. 20 shows that the yield criterion of Bier et al. [23, 24] is applicable as well for PVC foams, although both yielding and frac-

495

ture of the specimen are involved, i.e. yielding for pure compression and combined compression-torsion loading and fracture for pure torsion, pure tension 38

ACCEPTED MANUSCRIPT

and combined tension-torsion loading. The failure surface is evidently shifted towards the axis of the positive principal stress invariants, i.e. tensile loading. This feature was proven also by other authors dealing with failure surfaces of PVC foams [12, 13], whereas other

CR IP T

500

polymeric foams, e.g. PMI foams, display a less pronounced shifting [15]. The

shifting of the failure surface is determined by the difference between failure limits for tension and compression. More specifically, it depends on the distinct mechanical behaviour of PVC foams subjected to tension or compression, 505

i.e. pseudoelastic with final brittle fracture for tensile loading and pseudoelas-

AN US

tic with peak yield stress and protracted plateau for compressive loading. The

chosen failure criterion proved its suitability as well for failure data, which exhibit shifting. Given the parameters summarised in Table 4, the failure surface can be extrapolated towards the axis of negative first principal invariants, i.e. 510

beyond failure data points of uniaxial compression loading for the sake of eliminating the aforementioned shifting. The extrapolated failure surface is depicted

ED

triaxial compression.

M

in Fig. 20 and, according to Shafiq et al. [12], it concerns the field of biaxial and

4. Conclusions

In the present contribution different uniaxial and multiaxial tests have been

515

PT

R executed on the Divinycell HP100 PVC foam. Tension, compression and tor-

sion as well as combined tension-torsion and compression-torsion have been

CE

applied on cylindrical specimens for gathering failure data and the subsequent outlining of the failure surface in the invariants plane. In line with the work of

520

Jung and Diebels [20], the yield function of Bier et al. [22, 23, 24] has been suc-

AC

cessfully adapted to the collected failure data of the PVC foam. Furthermore, the occurrence and the evolution of deformation bands has been monitored for each loading case by a 3D-DIC method, which was performed through an 8-camera system. The above-mentioned experimental set-up has allowed the

525

monitoring of almost 360◦ around the axis of the specimen. Aiming to the in-

39

ACCEPTED MANUSCRIPT

spection of the whole surface has been crucial for understanding the growing of deformation bands based on the loading case, as well as for identifying possible irregularities, e.g. misalignments during testing. Especially under torsional

530

CR IP T

loading, the observation of 360◦ of the specimen is needed to provide conclusive analyses of the formation of deformation bands. The 8-camera system has

been essential in any loading cases where compression or torsion are involved, because local strain fields were different along the four wedges, which were

observed by the respective cameras pair. Deformation bands for compression

loading are influenced by buckling. For torsional loading they are affected by

the intrinsic orthotropy of the foam. Specimens were manufactured from panels

AN US

535

having an elongation of cells towards the through-thickness direction. Hence, it is likely that the orthotropy of panels affected the positioning of deformation bands during torsion. Since the axis of the specimen is perpendicular to the direction of elongated cells, both the longest and the shortest sizes of the cells 540

may be found along the circle of the cylindrical specimen surface. Therefore,

M

it can be deduced that struts from cells with different sizes would deform by different magnitudes. It would cause the positioning of deformation bands in

ED

two diametrically opposite wedges of the specimen. This feature is found also in the case of torsion-superposition in tensile-preloaded specimens. Whereas, 545

it appears diminished in compression-preloaded specimens. The present work

PT

aimed to give a deeper insight on the failure mechanism of PVC foams under

CE

multiaxial loads.

5. Acknowledgements The authors thank Prof. Ing. Francesco Aymerich and Prof. Ing. Antonio

Baldi for fruitful discussions and for providing the PVC foam panels and the

AC 550

University of Cagliari (Universit`a degli Studi di Cagliari, Italy) for the financial support.

40

ACCEPTED MANUSCRIPT

References References [1] Stewart R. At the core of lightweight composites. Reinforced Plastics 2009;53(3):30–5.

CR IP T

555

[2] Herbeck L, Kleineberg M, Sch¨ oppinger C. Foam cores in RTM structures:

manufacturing aid or high-performance sandwich? In: 23rd SAMPE Europe International Conference. Paris 2002,.

[3] Zhang S, Dulieu-Barton J, Fruehmann R, Thomsen OT. A methodology for

AN US

560

obtaining material properties of polymeric foam at elevated temperatures. Experimental Mechanics 2012;52(1):3–15.

[4] Taher ST, Thomsen OT, Dulieu-Barton JM, Zhang S. Determination of mechanical properties of PVC foam using a modified arcan fixture. Composites Part A: Applied Science and Manufacturing 2012;43(10):1698–708.

M

565

[5] Gdoutos E, Daniel I, Wang KA. Failure of cellular foams under multiaxial loading. Composites Part A: Applied Science and Manufacturing

ED

2002;33(2):163–76.

[6] Burman M, Zenkert D. Fatigue of foam core sandwich beams-1: undamaged specimens. International Journal of Fatigue 1997;19(7):551–61.

PT

570

[7] Toftegaard H, Lystrup A. Design and test of lightweight sandwich T-joint

CE

for naval ships. Composites Part A: Applied Science and Manufacturing 2005;36(8):1055–65.

AC

[8] Sadler RL, Sharpe M, Panduranga R, Shivakumar K. Water immersion

575

effect on swelling and compression properties of Eco-core, PVC foam and balsa wood. Composite Structures 2009;90(3):330–6.

[9] Siriruk A, Weitsman YJ, Penumadu D. Polymeric foams and sandwich composites: Material properties, environmental effects, and shear-lag modeling. Composites Science and Technology 2009;69(6):814–20. 41

ACCEPTED MANUSCRIPT

580

[10] Thomsen OT. Sandwich materials for wind turbine blades present and future. Journal of Sandwich Structures & Materials 2009;11(1):7–26.

Assembly. Univ Loughb 2010;.

CR IP T

[11] Hogg P. Wind Turbine Blade Materials, Supergen Wind Phase 1 Final

[12] Shafiq M, Ayyagari RS, Ehaab M, Vural M. Multiaxial yield surface of transversely isotropic foams: Part II Experimental. Journal of the Me-

585

chanics and Physics of Solids 2015;76:224–36.

[13] Christensen RM, Freeman DC, DeTeresa SJ. Failure criteria for isotropic

AN US

materials, applications to low-density types. International Journal of Solids and Structures 2002;39(4):973–82. 590

[14] Altenbach H, Kolupaev VA. Classical and non-classical failure criteria. In: Failure and damage analysis of advanced materials. Springer; 2015, p. 1–66. [15] Fahlbusch NC, Becker W, Kolupaev VA, Geertz G.

Nonlinear mate-

595

ED

2016;227(11):3113–21.

M

rial behaviour and failure of closed-cell polymer foams. Acta Mechanica

[16] Benderly D, Putter S. Characterization of the shear/compression failure envelope of Rohacell foam. Polymer Testing 2004;23(1):51–7.

PT

[17] Hoo Fatt MS, Jacob AJ, Tong X, MacHado-Reyes A. Crushing behavior and energy absorption of PVC foam: an anisotropic visco-elastic-plastic-

CE

damage model. 21st International Conference on Composite Materials, Xian, 20-25th August 2017;.

600

AC

[18] Donnard A, Gu´erard S, Mah´eo L, Viot P, Rio G.

Multiaxial experi-

ments with radial loading paths on a polymeric foam. Polymer Testing 2018;67:441–9.

[19] Ayyagari RS, Vural M. Multiaxial yield surface of transversely isotropic

605

foams: Part I Modeling. Journal of the Mechanics and Physics of Solids 2015;74:49–67. 42

ACCEPTED MANUSCRIPT

[20] Jung A, Diebels S. Microstructural characterisation and experimental determination of a multiaxial yield surface for open-cell aluminium foams. Materials & Design 2017;131:252–64. [21] Jung A, Diebels S.

Yield surfaces for solid foams:

on experimental characterization and modeling. 2018;41(2):e201800002.

A review

CR IP T

610

GAMM-Mitteilungen

[22] Bier W, Hartmann S. A finite strain constitutive model for metal pow-

der compaction using a unique and convex single surface yield function. European Journal of Mechanics-A/Solids 2006;25(6):1009–30.

AN US

615

[23] Bier W, Dariel M, Frage N, Hartmann S, Michailov O. Die compaction of copper powder designed for material parameter identification. International Journal of Mechanical Sciences 2007;49(6):766–77.

[24] Hartmann S, Bier W. High-order time integration applied to metal powder plasticity. International Journal of Plasticity 2008;24(1):17–54.

M

620

[25] Wu Y, Qiao D, Tang L, Liu Z, Liu Y, Jiang Z, et al. Global topology of

ED

yield surfaces of metallic foams in principal-stress space and principal-strain space studied by experiments and numerical simulations. International Journal of Mechanical Sciences 2017;134:562–75. [26] Zhang X, Tang L, Liu Z, Jiang Z, Liu Y, Wu Y. Yield properties of closed-

PT

625

cell aluminum foam under triaxial loadings by a 3D Voronoi model. Me-

CE

chanics of Materials 2017;104:73–84. [27] Luo G, Xue P, Sun S. Investigations on the yield behavior of metal foam

AC

under multiaxial loadings by an imaged-based mesoscopic model. Interna-

630

tional Journal of Mechanical Sciences 2018;142-143:153–62.

[28] Rajput MS, Burman M, K¨oll J, Hallstr¨ om S. Compression of structural foam materials–experimental and numerical assessment of test procedure and specimen size effects. Journal of Sandwich Structures & Materials 2019;21(1):260–88. 43

ACCEPTED MANUSCRIPT

635

[29] Miyase A, Wang SS. Test method development and determination of threedimensional stiffness properties of polyvinyl chloride structural foams. Journal of Composite Materials 2018;52(5):679–88.

CR IP T

[30] Bogusz P, Gieleta R, Konarzewski M, Stankiewicz M. Crushing behaviour

of the PVC foam loaded with beaters of various shapes. Acta Mechanica et Automatica 2018;12(1):23–30.

640

[31] Ye N, Zhang W, Li D, Huang W, Xie W, Huang X, et al. Dynamic response and failure of sandwich plates with PVC foam core subjected to impulsive

[32] DIAB .

R HP. Divinycell

AN US

loading. International Journal of Impact Engineering 2017;109:121–30. 2017.

http://www.diabgroup.com/en-GB/

Products-and-services/Core-Material/Divinycell-HP, 2018-04-16.

645

[33] DIAB .

R Divinycell H.

2018.

http://www.diabgroup.com/en-GB/

Products-and-services/Core-Material/Divinycell-H, 2018-05-31.

M

[34] Colloca M, Dorogokupets G, Gupta N, Porfiri M. Mechanical properties and failure mechanisms of closed-cell PVC foams. International Journal of Crashworthiness 2012;17(3):327–36.

ED

650

[35] Luong DD, Pinisetty D, Gupta N. Compressive properties of closed-cell

PT

polyvinyl chloride foams at low and high strain rates: Experimental investigation and critical review of state of the art. Composites Part B:

CE

Engineering 2013;44(1):403–16. 655

[36] Gdoutos EE, Daniel IM, Wang KA. Multiaxial characterization and model-

AC

ing of a PVC cellular foam. Journal of Thermoplastic Composite Materials 2001;14(5):365–73.

[37] ASTM D1623-09 . Standard test method for tensile and tensile adhesion

660

properties of rigid cellular plastics. Tech. Rep.; ASTM; 2009.

[38] Dantec Dynamics GmbH . Istra 4D Software Manual Q-4xx System. Manual Version: 2.8.5. Software Version: 4.4.4.459 2016;. 44

ACCEPTED MANUSCRIPT

[39] University of Colorado Boulder . Review of continuum mechanics: field equations. 2018. http://www.colorado.edu/engineering/cas/courses. d/NFEM.d/NFEM.Ch08.d/NFEM.Ch08.pdf, 2018-09-21. [40] Gibson LJ, Ashby MF. Cellular solids: structure and properties. Cambridge

CR IP T

665

University Press; 1990.

[41] Ashby MF, Evans T, Fleck NA, Gibson L, Hutchinson J, Wadley H. Metal foams: a design guide. Butterworth-Heinemann; 2000.

[42] Bert C, Reddy J. Mechanics of bimodular composite structures. In: Me-

AN US

chanics of Composite Materials: Recent Advances. Elsevier; 1983, p. 323–

670

37.

[43] Barsotti M, Puryear J, Stevens D. Developing improved civil aircraft ar-

AC

CE

PT

ED

M

resting systems; vol. 29. Transportation Research Board; 2009.

45