Examination of the failure of sandwich beams with core junctions subjected to in-plane tensile loading

Composites Science and Technology 69 (2009) 1447–1457

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Examination of the failure of sandwich beams with core junctions subjected to in-plane tensile loading M. Johannes *, J. Jakobsen, O.T. Thomsen, E. Bozhevolnaya Department of Mechanical Engineering, Aalborg University, Pontoppidanstrde 101, 9220 Aalborg East, Denmark

a r t i c l e

i n f o

Article history: Received 19 February 2008 Received in revised form 12 August 2008 Accepted 7 September 2008 Available online 18 September 2008 Keywords: C. Sandwich structures Core junctions C. Stress concentrations C. Failure criterion B. Fatigue

a b s t r a c t The paper concerns local effects occurring in the vicinity of junctions between different cores in sandwich beams subjected to tensile in-plane loading. It is known from analytical and numerical modelling that these effects display themselves by an increase of the bending stresses in the faces as well as the core shear and transverse normal stresses at the junction. The local effects have been studied experimentally to assess the influence on the failure behaviour both under quasi-static and fatigue loading conditions. Typical sandwich beam configurations with aluminium and glass-fibre reinforced plastic (GFRP) face sheets and core junctions between polymer foams of different densities and rigid plywood or aluminium were investigated. Depending on the material configuration of the sandwich beam, premature failure accumulating at the core junction was observed for quasi-static and/or fatigue loading conditions. Using Aluminium face sheets, quasi-static loading caused failure at the core junction, whereas no significance of the junction was observed for fatigue loading. Using GFRP faces, a shift of the failure mode from premature core failure in quasi-static tests to face failure at the core junction in fatigue tests was observed. In addition to the failure tests, the sandwich configurations have been analysed using finite element modelling (FEM) to elaborate on the experimental results with respect to failure prediction. Both linear modelling and nonlinear modelling including nonlinear material behaviour (plasticity) was used. Comparing the results from finite element modelling with the failure behaviour observed in the quasi-static tests, it was found that a combination of linear finite element modelling and a point stress criterion to evaluate the stresses at the core junction can be used for brittle core material constituents. However, this is generally not sufficient to predict the failure modes and failure loads properly. Using nonlinear material properties in the modelling and a point strain criterion improves the failure prediction especially for ductile materials, but this has to be examined further along with other failure criteria. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Structural sandwich elements represent a special type of layered composite elements comprising two relatively thin, stiff and high strength outer layers, the face sheets, which are adhesively bonded to a relatively thick, compliant and lightweight inner layer, the core. The resulting sandwich element forms a lightweight, stiff and strong structure which outperforms mass-equivalent monolithic structures for most load cases. However, problems arise when local effects disturb the ideal distribution of loads in the components of the layered sandwich. These local effects may occur due to discontinuities such as changes of geometry or material properties or when localised external loads are applied. It is wellknown that in the vicinity of sandwich sub-structures, e.g., joints, stiffeners or inserts, large stress concentrations are inevitably pres-

* Corresponding author. E-mail address: [email protected] (M. Johannes). 0266-3538/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2008.09.012

ent. These may initiate local failure processes (crack initiation) which may lead to global failure of the whole structure [1]. The local effects that are caused by the mismatch of elastic properties of adjoining materials in a sandwich core junction display themselves by local face sheet bending at the junction in combination with a rise of the in-plane stresses in the sandwich faces as well as of the shear and through-the-thickness stresses in the adjacent cores [2]. Depending on the type of loading and the configuration of the sandwich structure, the stress concentrations may cause local fracture of the core materials, but local face failure or interface failure are also possible scenarios. The influence of such local effects on the structural integrity has been studied under both quasi-static and fatigue loading conditions for sandwich structures subjected to transverse shear loading in [3,4]. The present paper focuses on the local effects induced at a junction between two different core materials in a sandwich beam subjected to tensile in-plane loading, a secondary load case which also occurs commonly in practice. A preliminary investigation of this was presented by Bozhevolnaya et al. [5,6]. In the present paper,

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the influence of the local effects on the failure behaviour under quasi-static as well as fatigue loading conditions is studied for five typical sandwich configurations. In the quasi-static tests, the specimens are subjected to loading up to failure. The failure loads and failure modes will be discussed and elaborated on based on results obtained from linear and nonlinear finite element modelling (FEM). In the fatigue tests, the focus is on observation of failure initiation and development. For two sandwich configurations, the fatigue life of the specimens will be compared to that of a set of reference specimens without a core junction. 2. Prediction of failure in sandwich structures A large number of studies concerning the failure behaviour of sandwich structures as well as the material and failure behaviour of sandwich constituent materials have been published in recent years. The general failure modes of sandwich structures under various static load cases have been described in textbooks [1,7] or in more recent studies [8,9]. For out-of plane loading, also the failure due to local effects at core junctions has been addressed by various studies [3,4,10], and for the special case of rigid inserts there are some design guidelines available for design engineers [11]. However, for the in-plane loading case the focus has generally been on sandwich structures loaded in compression, and the corresponding failure modes such as compressive face sheet failure, core shear instability, face sheet wrinkling or global buckling of the sandwich. The failure associated with local effects at core junctions in sandwich structures subjected to in-plane loading has received little attention. To the knowledge of the authors, only one study carried out parallel to this one has covered that topic [12]. Furthermore, a reliable and physically based failure criterion to be used in connection with the predicted stress concentrations at core junctions is still missing. The effects of core junctions have been examined for bi-material butt junctions in [10], where a point stress criterion basing on the first principal stress evaluated at a characteristic distance from stress singularities has been found to describe the failure behaviour well. However, in that study only brittle PMI foam materials, which behave linear elastically almost over the whole range of deformations until failure, were used, and the stress prediction could be done with linear FEM. In general, for failure prediction, knowledge about the deformation and failure behaviour of the constituent materials is crucial. The relevant material data can relatively easy be obtained for most face sheet materials (except for compression failure of FRP), but proper data for the core materials are more difficult to determine, as many of them show strongly nonlinear load-deformation behaviour. Material suppliers generally provide only linear elastic properties, and tend furthermore to state conservative minimum ultimate stress and strain values rather than actual mean values or even tested data from each production batch. For some materials, such as PVC foam cores, there can be significant scatter in the published material properties, making failure prediction even more difficult. Moreover, many sandwich core materials, such as honeycombs, wood and certain foam core materials, display anisotropic material behaviour. This adds complexity both to materials testing as well as to the establishing of accurate and reliable failure criteria. For cellular sandwich core materials, several studies have treated the failure characterisation taking multiaxial stress states into account [13–15]. A capped Tsai-Wu criterion was used with success for failure prediction. However, as mentioned before, reliable data on the material behaviour is sparse, and the data provided in [13–15] required extensive material testing and covered only very few of the available polymer foam core materials. Thus, in the present paper simple stress or strain based failure criteria are used.

3. Test specimens Five different sandwich beam configurations have been used to examine the influence of the local effects caused by core junctions in practice. Two designs of dog bone shaped specimens as shown in Fig. 1 were used in the experiments. Each specimen comprised four core sections, and the dog bone shape was used to ensure that failure is confined to a specific gauge section of the specimen. The gauge section is the middle part of the specimen with nominal width w, where two different core materials form a so-called butt junction. At the ends of the specimen, plywood or aluminium was used as an edge stiffener for a proper load introduction. The specimen configurations are specified in Table 1, where hf and hc denote the face and core thicknesses, respectively, and nstatic and nfatigue are the number of specimens used for the quasi-static and fatigue testing, respectively. In the reference configuration 2c only one core material was used, i.e., there was no core junction in the gauge section. Two techniques were employed to manufacture the specimens of configurations 1a/b and 2a/b/c, respectively. For configurations 1a/b, aluminium strips were bonded to previously manufactured core layers. The core layers consisted of the two polymeric foam core materials as described in Table 1 and two aluminium blocks as edge stiffeners, bonded together. The bonding of the face sheets to the core layer was done such that there was adhesive only between the face sheets and the foam cores, whereas bolted fastenings joint the face sheets and the aluminium blocks, simply in order to allow for a reuse of the aluminium blocks. For all bonds, an AralditeÒ 2011 epoxy adhesive was used. To bond the face sheet strips to the core layer, the specimens were stacked and a uniform pressure applied to all specimens simultaneously. After curing of the resin, the specimens were machined to the dog bone shape. The specimens of configurations 2a/b/c were produced by liquid resin vacuum infusion. The lay-up of the face sheets consisted of two layers of a bidirectional stitched non-crimp fabric with an areal weight of 650 g/m2. A previously manufactured core plate was placed on the lay-up of the lower face, and the lay-up of the upper face was placed on top of the core layer. The core plate consisted of the two core materials as described in Table 1 and two

Fig. 1. Schematic of the dog bone shaped specimens used in the tensile tests; top: specimen for configuration 1a/b; bottom: specimen for configuration 2a/b/c.

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M. Johannes et al. / Composites Science and Technology 69 (2009) 1447–1457 Table 1 Considered sandwich beam configurations

1a 1b 2a 2b 2c

Face material

Core material 1

Core material 2

hf (mm)

hc (mm)

w (mm)

nstatic

nfatigue

Aluminium 7075-T6 Aluminium 7075-T6 GFRP, NCF [0/90]s GFRP, NCF [0/90]s GFRP, NCF [0/90]s

Divinycell H60 Rohacell 51WF Divinycell H60 Divinycell H200 Divinycell H60

Divinycell H200 Rohacell 200WF Divinycell H200 Plywood

0.4 0.4 1.0 1.0 1.0

25 25 25 25 25

10 10 20 20 20

3 3 3 6 –

5 – 8 5 4

plywood sections as edge stiffeners, bonded together. The complete lay-up was bagged and through a one-step vacuum infusion process a sandwich plate was produced, from which sandwich beams were cut out and then machined to the dog bone shape.

700

Stress [MPa]

600

4. Material properties

400 300 200 True stress-strain

100

Eng. stress-strain

0 0

2

4

6 8 Strain [%]

10

12

14

Fig. 2. Uniaxial stress–strain curve of aluminium; blue: engineering stress–strain; red: true stress–strain. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)

500 400 Stress [MPa]

The material data were partly taken from data sheets provided from the manufacturers, partly obtained by materials testing at the manufacturer and partly by in-house materials testing. The material data is essential for the numerical modelling of the sandwich specimens presented later on. An overview over the elastic and strength properties of the sandwich constituent materials is given in Table 2. The ultimate stress and strain values are given in terms of engineering stress and strain. The ‘‘characteristic distance” is a fracture mechanics property that will be used for the failure prediction later on. A description of the ‘‘characteristic distance” and the corresponding failure criteria will be provided in the finite element modelling section of this paper. The sandwich constituent materials have very different material behaviour. Where the GFRP is known to display approximately linear stress–strain behaviour over the whole range of deformations, the aluminium and the Divinycell foam cores show nonlinear material behaviour at higher strains. This will be taken into account in the nonlinear FEM. Nonlinear stress strain data for the H60 and H200 foam core materials were made available from DIAB AB [18], and data for the aluminium were obtained from in-house tensile tests according to EN 10002-1. Uniaxial stress–strain curves for the aluminium and the GFRP are given in Figs. 2 and 3, respectively, and stress–strain curves for the Divinycell H60 and H200 core materials are given in Figs. 4 and 5, respectively [18]. From the Rohacell materials it is generally known that the material behaviour is almost linear elastic until failure [21]. Fig. 6 shows the results of the tensile tests performed on the plywood and it can be seen that the behaviour of the plywood can also be approximated well by a linear curve fit. The stress strain curves resulting from the material testing are in engineering stress and strain. As the aluminium is exposed to

500

300 200 100 0 0

0.5

1

1.5 Strain[%]

2

2.5

3

Fig. 3. Uniaxial stress–strain curve of GFRP.

very high stress/strain levels due to large plastic deformation, the engineering stress and strain data were converted into true/natural stress and strain for the further use in the nonlinear FEM. The conversion can be justified by the fact that the aluminium does not undergo a volume change during deformation [22]. In Fig. 2 both the engineering and the true/natural stress and strain are plotted. For

Table 2 Material data of the sandwich constituent materials Material

E-modulus (MPa)

Poisson’s ratio [ ]

Yield/ultimate stress (MPa)

Ultimate strain (%)

Characteristic distance (mm)

Aluminium 7075-T6 GFRP, NCF, [0/90]sb

71700a 24200a

0.33 [16] 0.12c

480a/550a –/465a

12.70a 2.45a

– –

DIAB H60 (linear) DIAB H60 (nonlinear) Rohacell 51WF

60 [17] 37 [18]d 75 [19]

0.32 [17] 0.32 [17] 0.32 [20]

–/1.4 [17] –/1.2 [18] –/1.6 [19]

5.7 [18] 5.7 [18] 3.0 [20]

1.37

DIAB H200 (linear) DIAB H200 (nonlinear) Rohacell 200WF Plywood

310 [17] 359 [18]d 350 [19] 9200a

0.32 [17] 0.32 [17] 0.38 [20] 0.32e

–/4.8 [17] –/7.3 [18] –/7.0 [19] –/53.0a

4.7 [18] 4.7 [18] 3.5 [20] 0.63a

a b c d e

In-house material testing. Fibre volume fraction approx. 50% (determined by burn tests and by weighing). Calculated with classical laminate theory. Initial E-modulus. Simplifying assumption.

0.62 1.16 0.38 0.60e

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1.4

Stress [MPa]

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

1

2

3 Strain [%]

4

5

6

Fig. 4. Uniaxial stress–strain curve of DIAB H60 [18].

8

Stress [MPa]

7 6 5 4 3 2 1 0 0

1

2

3 Strain [%]

4

5

6

Fig. 5. Uniaxial stress–strain curve of DIAB H200 [18].

60

Stress [MPa]

50 40 30 20 10 0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Strain [%] Fig. 6. Uniaxial stress–strain curve of Plywood; blue: test results; red: linearisation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)

the Divinycell materials the assumption of a constant volume during large deformations is not meaningful because of the mechanics of cellular polymer materials (yielding depends on both deviatoric and hydrostatic stresses). In fact, it is not clear to which extent the nonlinear behaviour observed in the materials testing results from plasticity, nonlinear elasticity or some combination of nonlinear (visco)elastic-plastic behaviour. To the knowledge of the authors there is no common agreement about which material model is appropriate to model the nonlinear behaviour of Divinycell PVC foams. On the account of this the material curve for the nonlinear FEM was simply formulated in engineering stresses and strains as measured in the material tests for the H60 and H200. It has to be noted that for the linear FEM exclusively the linear elastic values that are mostly from data sheets as given in Table 2 will be used as material input, whereas in the nonlinear model the whole stress–strain curves as shown in Figs. 2–5 will be considered. The experimentally obtained curves are fitted by piece-wise linear curves as indicated by the markers in Figs. 2–5. It should also be noted that those nonlinear material data represent exemplary

stress–strain curves from the materials tests at the manufacturer, and their Young’s modulus and ultimate stress show some deviation from the datasheet values. For both the linear and nonlinear modelling all materials were assumed to be isotropic, which is a substantial simplification for the GFRP face sheets and the plywood. However, it is considered as reasonable for the face sheets, as the nominal stress state in the faces of a sandwich beam is that of a membrane with in-plane normal stresses in the longitudinal direction dominating. For a narrow sandwich beam section the stress state in the face sheets can be approximated as a plane stress state (close to a unidirectional stress state), as will be discussed in more detail in Section 6. This means that the assumption of isotropic material properties of the GFRP is justified for the areas of the sandwich away from the core junction where the nominal stress state is dominant. In the area of the local effects near the junction the local stresses need special consideration and a layered model of the composite may be more accurate. The assumption of isotropy causes inaccuracies for two reasons. The first one is the error due to erroneous/inaccurate material properties in the through-thickness direction. However, even in the area of local effects the through-thickness stresses are very small compared to the in-plane stresses, so that the deviation of the presumed material data compared to the real throughthickness data does not have a big influence. The second and most important source of inaccuracies is the difference in the flexural stiffness of a material model assuming isotropy and that of the real layered material. For the given lay-up the isotropic material will have a lower flexural stiffness as compared to the real layered material. On the other hand, there is a resin rich layer on each side of the GFRP laminate of the specimens, which in turn reduces the difference in the flexural stiffness and thereby the inaccuracies in stiffness predictions associated with the assumption of isotropy. As the overall/global stiffness of the face sheets is modelled accurately and the local stiffness is modelled reasonably well, the influence on the core stresses should be minor. Accordingly, the error made by the simplified material model is believed to be acceptable in all cases where the failure of the sandwich is driven by the core material. The resulting stresses in the face sheets would obviously be different on the layer/lamina level for a layered model. However, a layered model would not be more accurate with respect to the strength of the GFRP faces, as the strength of the GFRP was measured as an overall strength of the laminate, and no strength data are available for the single layers. For the plywood core, there is no such justification for a simplified material model, as there are both transverse and throughthickness stresses present in a sandwich core, but as no multiaxial data were available for the plywood, isotropic material behaviour was assumed for simplicity. In this case, a simplifying assumption was also made for the value of the Poisson’s ratio. Due to their manufacturing process the DIAB materials show a slightly anisotropic behaviour as well, with a somewhat higher stiffness and strength and lower ultimate strain in the through-thickness direction than in the in-plane directions. This is stated by the manufacturer and has been shown in independent tests [13–15]. However, a full multiaxial characterisation of the material was not available for the types of foam used in this study, and thus isotropic material behaviour was assumed for simplicity. The Rohacell foams are produced with a different process and are practically completely isotropic so that they can be described well by an isotropic linear elastic material model. 5. Experimental investigation – part 1: quasi-static tests Quasi-static tests to failure were carried out to assess the influence of the local effects on the static failure behaviour and the failure load. The tests were carried out on a Schenck Hydropuls

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Fig. 7. Experimental set-up with double-hinged loading fixture.

5000

Core + face failure at junction

4500 4000

Final face failure at junction Core failure at junction

Load [N]

3500 3000 2500 2000 1500 1000

Configuration 1a Configuration 1b

500 0 0

1

2

3 4 5 6 Displacement [mm]

7

8

9

Fig. 8. Load–displacement curves of tests with configurations 1a and 1b.

20000

Face failure away from junction

18000

Face failure at junction

16000 14000 Load [N]

servohydraulic test machine, and a double-hinged fixture as shown in Fig. 7 was used to avoid bending and ensure a pure in-plane loading. Three or six specimens, respectively, of each configuration (see Table 1) were subjected to axial tensile loading until final failure. The tests were run in load-controlled mode (due to control irregularities observed in displacement-controlled mode) with a load rate of 0.1 kN/s, and the load and crosshead displacement data were recorded. One specimen of each configuration was equipped with strain gauges on the outer face surface on each side of the specimen, as indicated in Fig. 1. This was done to ensure there was no bending loading, and to allow for a rough correlation of the FEM strain data with the displacement and strain data from the experiment within the linear range of deformations. Video recordings and in some cases high-speed video recordings were taken during the tests. Figs. 8 and 9 show the distinctively different load–displacement curves of the tests with configurations 1a/b and 2a/b, respectively. Except from the differences in the failure loads the individual load–displacement curves of the specimens were very similar and very close to the average within each configuration, so that only one representative curve of each configuration is shown in Figs. 8 and 9. Configurations 1a/b correspond to a case where the face material is more ductile than the core, and in configurations 2a/b the opposite is the case. For all configurations, the load–displacement behaviour is dominated by the stiff face sheets. For configurations 1a/b failure was dominated by plastic deformation of the aluminium faces once their yield stress was reached, leading to failure of the compliant foam core at its ultimate strain and followed by final face failure. The core failure is indicated in Fig. 8 and appears for configuration 1b as a small load drop and/or displacement jump in the curve. After the first core failure the proceeding behaviour was dependent on the type of core. For configuration 1a the core failure was immediately accompanied by consecutive face failure, whereas for configuration 1b the face sheets continued to yield a little longer. This is because of the more ductile behaviour and therefore a higher ultimate strain of the DIAB foams compared to the brittle Rohacell foams (see Table 2). The Rohacell 51WF foam fails at a lower strain and load level where a redistribution of the load into the face sheets is still possible. At the higher ultimate strain of the DIAB H60 the load level is higher, the face sheets are in a more progressed stage of yielding and the ability for load

Core failure at junction

12000 10000 8000 6000 4000

Configuration 2a

2000

Configuration 2b

0 0

1

2

3 4 5 Displacement [mm]

6

7

8

Fig. 9. Load–displacement curves of tests with configurations 2a and 2b.

redistribution is limited. In all cases the core failure was visible by a crack opening in the compliant core almost parallel to the core junction. Two of three specimens of configuration 1a, and all specimens of configuration 1b failed in the vicinity of the junction. It is assumed that the stress concentrations at the junction may have caused a pronounced local face yielding in this area, leading to subsequent core failure and final failure close to the junction. Evidence for the local yielding influenced by the local effects is that yielding starts at a load of about 3700 N for both configurations, which corresponds to a nominal face stress of about 450 MPa. This is only about 94% of the nominal yield strength of 480 MPa, and an indication that in some areas the local face stress may indeed have reached 480 MPa at that load as a consequence of the local effects. The same applies also to the final failure load. With respect to the failure prediction discussed later in the paper, the course of failure described above indicates that nonlinear modelling is necessary to describe the failure behaviour accurately. It further indicates that for the first failure of the core a strain criterion may be a meaningful approach, and that in particular the first principal (crack opening) strain/stress deserve closer attention. From the load–displacement plot in Fig. 9 it can be seen that for configuration 2a/b the sandwich deformations, again reflecting the face sheet deformations, show roughly linear behaviour up to final failure when the ultimate strain of the GFRP of about 2.5% is reached. In this range of strains also the foam core material behaves almost linearly, which indicates that for configurations 2a/b the linear FEM may be sufficient to predict the failure behaviour. For configuration 2a, the core junction did not seem to have an influence on the failure behaviour, and the specimens simply failed by tensile face failure at arbitrary locations in the gauge section. For configuration 2b with the junction between plywood and H200, failure was observed in the core/core interface two times, it

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was observed in the H200 foam two times, and twice both the H200 and Plywood materials failed practically at the same time. Closer examination of the fractured interfaces showed that material of the other core could be found on both sides. For the cases of failure in the H200 foam high-speed video recordings showed that failure initiated at the tri-material corner of the core junction. For the cases of failure of both H200 and plywood high-speed video recordings were not available, and it cannot be concluded clearly where the failure initiated. However, it can be stated that the core junction had an influence on the failure behaviour. Table 3 gives a summary of the observed failure loads, failure modes and failure locations of the quasi-static tests. Fig. 10 shows images from video recordings showing the typical failure modes of each specimen configuration. As the failure occurred directly at or in the vicinity of the junction for several of the tested sandwich configurations it is clear that the local effects at the core junction/ discontinuity have an influence on the failure behaviour under in-plane tensile loading. As the configurations differ greatly in their deformation and failure behaviour, however, linear and nonlinear finite element modelling are used in the following to explain the failure behaviour as described above, as well as to propose criteria for failure prediction of sandwich beams with core junctions under in-plane tensile loading. 6. Finite element modelling Finite element modelling of sandwich beam sections with core junctions has been used to evaluate the local stresses and strains at the core junctions numerically. Both linear and nonlinear FEM modelling was used. The nonlinear modelling adopted geometrical nonlinearity (large deformations and rotations) as well as nonlinear material behaviour (e.g., material plasticity). Both two dimensional (2D) and a three dimensional (3D) models were used in the initial stage of the study. However, the analyses conducted later on were limited to 2D modelling for a number of reasons. It is known from previous strain gauge measurements conducted on similar specimens that the transverse deformation of the face sheets is practically unrestricted, so that for the faces a plane stress assumption is appropriate. For the core, different Poisson’s ratios of the face and the core may lead to transverse stresses close to the face/core interface. There may also be transverse stresses at the junctions of a compliant core to a stiffer core, so that in these regions the stress state may be closer to plain strain. However, the plain strain assumption would artificially increase the overall inplane stiffness of the structure and may lead to transverse stresses that are not present in reality. Further, the 3D modelling showed that the deviations from a plane stress condition were generally small, in particular for configurations 1a/1b with transverse stresses of magnitude less than 5% of the longitudinal or through-thickness stresses. For configuration 2b, no reliable multiaxial material data were available for the plywood to create a 3D model. Finally, there was a limitation to the number of elements in the FE program (ANSYS Version 10 with Academic Teaching Advanced license), so

that the element size in the 3D model was always much larger than for a 2D model, which in some cases could cause mesh dependent results and additional errors for the 3D model. Thus, a 2D model using a plane stress assumption was preferred. The material combinations and face and core thicknesses were chosen in accordance with the sandwich configurations used in the experiments (see Table 1). Isoparametric two-dimensional 8node PLANE 183 elements were used for the finite element mesh of the 2D model. The mesh was refined at the core junction to obtain sufficiently small element edge lengths that ensure convergence of the results. The resulting element size adjacent to the core junction was 1/32 mm for the linear model, and 1/8 mm for the nonlinear model in order to reduce the computational effort to a reasonable level. The bond between the face sheets and the core and between the two core materials at the junction was assumed to be perfect and very thin, and hence no additional layers of adhesive were considered in the analysis. Fig. 11 shows a sandwich section with a core junction and the corresponding finite element mesh at the tri-material corner of face sheets and cores. The deformation pattern indicated in Fig. 11 is an idealisation of the real face and core deformation due to Poisson’s effects, i.e., the different longitudinal extension of the cores due to their different stiffness leads to different through-thickness strains. Instead of modelling the entire sandwich specimen with the load being introduced via the rigid outer core sections, a 200 mm long sandwich section with the tri-material corner as indicated in the left of Fig. 11 was modelled. Symmetry with respect to the y–z-plane (see Figs. 1 and 11) was chosen as the boundary condition for the beam end with the compliant core, and tensile pressure loads in accordance with the nominal stresses were applied on the face and the core at the other end of the beam, as indicated in Fig. 11. The symmetry boundary condition is not used as a geometric representation of the real structure, but it simply corresponds to a zero displacement in the x-direction and a constraint of rotation around the z-axis. For the linear model only the elastic material properties as given in Table 2 were used as material data, whereas for the nonlinear model the piece-wise linear stress strain curves given in Figs. 2–5 were input into the model. For the linear model the loads were chosen to be equal to the failure loads observed in the quasi-static tests, but results can be scaled as it is a linear analysis. For the nonlinear model, the loads were incrementally increased by the FE program until convergence of the results was not achieved any more. Non-convergence can generally occur for several reasons. In this case it indicated that the stresses/strains of one of the nonlinear materials approached the ultimate stress/strain, and the final part of the stress–strain curve with very little incremental stiffness was reached. That point marks the load bearing capacity for that material and the first failure in the structure. However, neither voids nor cracks were created in the model when one of the materials exceeded its ultimate stress/strain, and the stress level for that material remained constant. Thus both the linear and nonlinear modelling will only be strictly meaningful to simulate failure initiation, whereas the following

Table 3 Average failure loads, failure modes and failure location in the quasi-static tests First failure load (N) ± SD

Location of first failure (distance from junction)

1a 1b

=Final failure 4085 ± 375

Compliant core, at junction (4 mm, >20 mm, 2.5 mm) Compliant core, at junction (15 mm, 5.5 mm, 7 mm)

2a 2b

=Final failure 11801 ± 742

=Final failure Compliant/stiff core/interface (0 mm for all specimens)

Final failure load (N) ± SD 4444 ± 06 4262 ± 86 17831 ± 1195 16392 ± 879

Location of final failure (distance from junction) Face sheet, at junction (8 mm, >20 mm, 1 mm) Face sheet, at junction (0 mm, 12.5 mm, 8.5 mm) Face sheet, arbitrary locations (>20 mm) Face sheet, at junction (0 mm for all specimens)

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Fig. 10. Pictures of typical failure modes in the quasi-static tests.

Fig. 11. Sketch of a sandwich core junction and the FE mesh at the tri-material corner.

570

Face Surface Face/Core Interface

Stress in faces σx [MPa]

560 550

Nominal stress σx H60: 541.34 MPa

540 530

Nominal stress σx

520

H200: 489.38 MPa

510 500 490 480 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 Distance across junction [mm]

Fig. 12. Face stresses for sandwich configuration 1a at the failure load (linear model).

is associated with a local rise of the face normal stresses (see Fig. 12) and the core shear and through-thickness normal stresses (see Fig. 13). It has to be noted that as the tri-material corner is inherently associated with a stress singularity within the framework of linear elasticity [23], the core stresses predicted by the linear model are infinitely high at the tri-material corner and thus the path for evaluation of the core stresses cannot be located directly at the face/ core interface. Thus, for the linear model the core stresses are assessed at a characteristic distance away from the interface. The approach of a characteristic distance was proposed by Ribeiro-Ayeh, Hallström and Grenestedt together with a point stress criterion to examine the failure of brittle sandwich foam core materials at bimaterial interfaces [10] and of fracture of cracks and wedge shaped notches in PVC foam [24]. According to [10,24], the characteristic distance is assumed to be a material constant which can be calculated from the fracture toughness and the ultimate stress of a material as follows:

Core stresses close to interface [MPa]

progressive development of plastic yielding and finally failure cannot be predicted accurately by the adopted approach. The results of the FEM were analysed in various ways. First, the predicted face stresses along the face outer surfaces, along the face/ core interfaces and the core stresses were read out on various paths along the length direction (x-axis) of the sandwich beams to obtain an impression about the characteristics of the local effects. Figs. 12 and 13 show the face and core stresses, respectively, as calculated by the linear model for sandwich configuration 1a at the experimentally observed average failure load. The stresses rx, ry and sxy denote the normal stresses and the shear stress in the global x–y coordinate system, as shown in Figs. 1 and 11, and r1 and r2 denote the first and second principal stresses, respectively. The plots show the characteristics of the local effects at the junction. The larger contraction of the compliant core than of the stiff core under in-plane tensile loading, and the necessary redistribution of stresses due to the stiffness mismatch, leads to local bending of the face sheets at the core junction (see Fig. 11) which

2.5 2 1.5

Nominal stress σx H200: 2.116 MPa

1 Nominal stress σx H60: 0.453 MPa 0.5 0 -0.5 -1

σx σy τxy σ1 σ2 -1.5 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 Distance across junction [mm]

Fig. 13. Core stresses for sandwich configuration 1a at the failure load (linear model).

1454

Rch ¼

M. Johannes et al. / Composites Science and Technology 69 (2009) 1447–1457

 2 1 K IC 2p r1

ð1Þ

The values for the characteristic distance in Table 2 were calculated using material data given in [10,21,24]. As an alternative approach, following the work by Bozhevolnaya et al. [5,6], the characteristic distance can be related to the microstructure of the material. In this case, as failure in the core material occurred in the compliant core, the characteristic distance was chosen as the average cell size acell = 0.6 mm [5,6] of the H60 foam. Both approaches were compared for the failure prediction. In case of the Rohacell 51WF foam, the characteristic distance equals to about 0.6 mm in both cases. As no data were available for the plywood, a value of 0.6 mm was used here as well for simplicity. For the failure prediction it was decided to use a point stress criterion, tying up on the work of Ribeiro-Ayeh, Hallström and Grenestedt [10,24], Bozhevolnaya et al. [5,6] and Jakobsen et al. [12]. For the nonlinear model the first principal stress and first principal strain were used; for the linear model the first principal stress was considered as well as a ‘‘simplistic” ultimate strain criterion. The analysis procedure was to read out the stresses of the core components into a so-called element table. For the linear model the elements within the characteristic distance around the tri-material corner were excluded hereby. From the element tables the maximum stress or strain could easily be read out. The location of the corresponding element was then checked in a contour plot. Table 4 gives an overview of the stresses in each sandwich constituent at the failure loads predicted by the different failure criteria using the stress results of the linear FE model. The bold entries in the table indicate the component that reaches its ultimate stress first and is predicted to fail first. Table 5 shows an estimation of the failure load by only considering the nominal stresses far away from the core junction, i.e., without considering the local effects. It also shows an estimation based on a ‘‘simplistic” ultimate strain criterion. This criterion is based on the nominal state as well, but gives the failure load according to the component reaching its ultimate strain first. Here the nonlinear stress–strain data (see Figs. 2–5) were used for reading out the stresses and calculating the failure load. As an example, for configuration 1a/b the stresses in the face sheet are higher in the section with the compliant core than in the stiff core and thus face yielding will occur in this section. The compliant cores fail first as their ultimate strains of 5.7% and 3.0% (H60 and 51WF), respectively, are lower than that of the aluminium. From the material data of the H60/51WF foams a stress of 1.2 MPa/1.6 MPa, and from the data of the aluminium a stress of 535 Mpa/520 MPa (using the engineering stress/strain data) is read out. Using the geometry data of Table 1, and assuming equal strain in face sheets and core, the failure load for the first failure is then calculated to 4580 N/ 4560 N. The stress in the stiff core at that load can only be estimated, e.g., to about 2.5 MPa when assuming that no yielding has occurred in the specimen section with the stiff core. This phenomenological criterion reflects the real failure occurrence well,

but it does not consider the local effects or the multiaxial stress state in the yielding zone. The bold entries in the table indicate the component that reaches its ultimate stress/strain first and is predicted to fail first. From Table 5 it can be seen that the prediction using the nominal stresses gives a relatively accurate failure load for the final failure of all configurations where failure is dominated by the face sheets. This is clear when keeping in mind that the face sheets carry the major part of the load and that the stress concentration in the faces at the junction is much less than the stress concentration in the core (see Figs. 12 and 13). However, it overestimates the loads for the first failure for 1a/b and 2b, as it does not account for the local effects. For configuration 2a the predicted failure load is very close, but the prediction for the failure location is not correct. The shortcomings of the nominal stress criterion are that it neglects the local effects and the nonlinear material behaviour that can lead to major discrepancies when the local effects are important as for configuration 2b. The ‘‘simplistic” ultimate strain criterion has the advantage that the material-wise nonlinearity is taken into account, and thus the prediction for the location of the first failure is better for configuration 1a/b and 2a. However, the failure loads are overestimated as the local effects are neglected, leading to a particularly big discrepancy for configuration 2b. From Table 4 it can be seen that the maximum stress criteria using the stress data of the linear FEM fail as well to predict the failure for configurations 1a and 1b. The predicted failure load is lower as for the prediction from the nominal stresses, as it considers the local stress rise at the junction as seen in Figs. 12 and 13, but failure is predicted to occur in the face sheet as the material nonlinearity is not taken into account. For configuration 2a the maximum stress criteria overestimate the local effects and predict failure at a drastically reduced load due to the local effects. However, this was not the case in the experiments where only face failure was observed, possibly because local core yielding can relieve the stress concentrations at the junction. The results are practically the same with the different characteristic distances. For configuration 2b where the local effects caused premature failure in the experiments the failure is predicted correctly, and especially the failure load from the material dependent distance is close to the experimentally observed failure load. Thus, for the case of nearly linear and brittle material behaviour, as for configuration 2b, it can be concluded that a maximum principal stress criterion evaluated at a material dependent characteristic distance with stress data from linear FEM can be used for reasonably accurate failure prediction. However, it is not feasible to extrapolate results obtained for the case of a bi-material interface between two brittle materials, where the adopted material dependent characteristic distance is based on a fracture mechanics approach [10,24], to the case of a tri-material corner with more complex loading and nonlinear material behaviour. Table 6 presents the results of the nonlinear modelling. Here, the failure behaviour of configurations 1a and 1b is modelled much better, and the failure in the compliant core is pre-

Table 4 Failure loads and stresses predicted by linear FEM and point stress criteria Exp. failure load (N)

1a 1b 2a 2b

4444 4085 17831 11801

Material dependent distance

Distance acell = 0.6 mm

Predicted failure load (N)

Predicted failure load (N)

4323 4379 11921 10320

Maximum principal stresses (MPa) Face sheet

Compl. core

Stiff core

550 550 310 459

0.84 1.15 1.40 4.80

2.47 3.25 4.11 –

4323 4379 10406 8939

Maximum principal stresses (MPa) Face sheet

Compl. core

Stiff core

550 550 271 397

0.95 1.15 1.40 4.62

2.72 2.98 4.05 53.00

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M. Johannes et al. / Composites Science and Technology 69 (2009) 1447–1457

Table 5 Failure loads and stresses predicted from nominal stresses and ‘‘simplistic” strain (*the nominal face stresses refer to the stresses in the sandwich section with the compliant core) Exp. failure load (N)

Prediction from nominal stresses Predicted failure load (N)

1a 1b 2a 2b a

4444 4085 17831 11801

4515 4544 17388 17388

Prediction from ‘‘simplistic” ultimate strain

Nominal stresses (MPa)

Predicted failure load (N)

Face sheeta

Compl. core

Stiff core

550 550 422 375

0.46 0.58 1.05 4.80

2.15 2.41 4.80 28.73

4580 4560 19000 21600

Nominal stresses (MPa) Face sheet

Compl. core

Stiff core

535 520 465 465

1.2 1.6 0.8 6.0

2.5 2.5 6.0 35.7

The nominal face stresses refer to the stresses in the sandwich section with the compliant core.

Table 6 Failure loads and stresses/strains predicted by nonlinear FEM and point stress/strain criteria Maximum principal stress criterion

1a 1b 2a 2b

4444 4085 17831 11801

4450 4250 17800 11800

Maximum principal strain criterion

Maximum principal stresses (MPa) Face sheet

Compl. core

Stiff core

540 475 450 310

1.3 1.4 1.3 6.3

4.3 3.4 6.9 69

dicted correctly. The failure loads are relatively close to the experimentally observed values, with a difference of 100 N between the maximum principal stress and the maximum principal strain criterion. The different results from the stress and strain criteria can be explained by the internal procedure of the FE program to calculate the stiffness matrix. With increasing load, the stiffness matrix is updated using the stress–strain data that was input as the material model, i.e., the stiffness matrix is dependent on the stress/strain level. However, if a multiaxial stress/strain state is present, the strains are converted into an equivalent strain to allow for a comparison with the uniaxial stress–strain data of the material model. In ANSYS, this is done according to the following relation (according to von Mises yield criterion):

e

1 ¼ 1 þ veff

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð1  2 Þ2 þ ð2  3 Þ2 þ ð3  1 Þ2 2

ð2Þ

where meff is the so-called effective Poission’s ratio

(

meff ¼

0

material Poisson s ratio for elastic strains 0:5 for plastic strains

ð3Þ

It is anticipated that this procedure can lead to stresses/strains in the model that are actually above the (uniaxial) ultimate stress/ strain of a material, as seen in Table 6. A certain tolerance has thus to be taken into account when interpreting the data. As an example, for configuration 2a the failure is predicted in the compliant core, but the stresses in the face and the stiff core are very close to the ultimate stress as well. According to the maximum principal strain criterion the failure is predicted in the face sheet as seen in the experiment. For configuration 2b both criteria predict failure in the stiff core. The major difference between the stress and the strain criterion is the location in each component where the failure is predicted to occur. For the stress criterion this is generally around the tri-material corner, whereas for configuration 1a/b the maximum strains are in the centre axis of the sandwich. Fig. 14 shows the core stresses and strains along the beam centre axis as calculated by the nonlinear model for sandwich configuration at the failure load. It can be seen that the ultimate strain is reached for the compliant H60 core, whereas the stresses are subcritical at the beam centre. Contrary, at the face/core interface the predicted stresses are determinative for the failure. Unfortunately it is not entirely clear from the experiments where the failure initiated and therefore it

Predicted failure

Maximum principal strains (%) Face sheet

Compl. core

Stiff core

4550 – 17900 11800

5.0 – 2.4 1.5

5.9 – 4.5 2.6

1.0 – 2.0 0.8

is not possible to make a final assessment about the criteria. It has also to be noted that the von Mises approach for calculation of the equivalent strain as given in Eqs. (2) and (3) is in general not suitable for cellular materials, as the deformation and failure behaviour of foams depends on both deviatoric and hydrostatic stresses. This may cause further inaccuracies in the calculation of the stresses close to the core junction due to regions with a hydrostatic stress state. As the stresses for the nonlinear model were evaluated directly at the face/core interface and not at a characteristic distance from the interface, the calculation of the equivalent strain and the effective stress as well as the available plasticity model and material data are the major factors of influence for the failure prediction in the nonlinear model. Overall, the nonlinear model captures the nonlinear material behaviour and failure behaviour much better than the linear model. However, it has to be concluded that some issues are still unclear and warrant further investigation. It is essential to consider the multiaxial stress/strain state in the core material. Even though substantial work has been done on the failure envelopes of some foam core materials, the data is still sparse and the integration into an FE model is not straightforward, in particular when considering the plasticity. Finally it has to be noted that the material properties of some of the used materials show substantial variations from one

7%

3.5

6%

3

5%

2.5

4%

2

3%

1.5

2% 1%

1

First principal strain First principal stress

0% -3 0 -25 -2 0 -15 -1 0 -5 0 5 10 15 Distance across junction [mm]

0.5

First principal stress [MPa]

Predicted failure load (N)

First principal strain [%]

Exp. failure load (N)

0

20

25

30

Fig. 14. Core stresses and strains in the centre axis of sandwich configuration 1a 1a at the predicted failure load.

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M. Johannes et al. / Composites Science and Technology 69 (2009) 1447–1457

production batch to another and thus limit the accuracy of any model. 7. Experimental investigation – part 2: fatigue tests The scope of the fatigue tests was to examine whether the local effects influence the failure behaviour under repetitive loading, and to compare the failure behaviour to that of the quasi-static tests. The tests were run with a sinusoidal load, a frequency of 3 Hz and a loading ratio R = 0.1. Table 7 gives an overview over the chosen fatigue loads and the resulting failure behaviour and fatigue life times. For configuration 1a, no failure could be observed in connection with the core junction. The chosen load was well below the regime where face yielding could occur and this might explain the different failure behaviour compared to static loading conditions. For configuration 2a, an accumulation of failure at the core junction was noticed, which had not been the case for quasistatic loading. During the fatigue tests, a gradual whitening of the face sheets was observed, the intensity of which was most pronounced at the core junctions, meaning that the local effects possibly led to an increased damage accumulation in this area. Fig. 15 shows images of the course of failure in the fatigue tests. Regarding the tests with configuration 2a to 2c it should be noted that a part of the specimens failed in the transition radii of the dog bones. As it is not clear whether this type of failure was a result of the transition or the core junction between the edge stiffener and the foam core (see Fig. 1) these specimens were not considered for evaluation. The total number of specimens of one specimen configuration is stated after the slash in Table 7, and the number of evaluated specimens is stated before the slash. A part of the tests with configuration 2a were run at different load levels, because an appropriate load level had to be established in the beginning. Altogether, the results clearly indicate that fatigue damage accumulation is connected to the stress concentrations at the core junction for sandwich configuration 2a.

For configuration 2b a similar course of failure as for configuration 2a was observed for one specimen that failed at the core junction (see Fig. 15). However, two specimens failed at arbitrary places in the beam gauge sections and two in the transition radii, thus preventing a solid conclusion due to the limited statistical significance. Configuration 2c was used as reference to compare with configurations 2a and 2b with respect to failure behaviour and fatigue life. Three specimens of configuration 2c failed in the transition radius and one specimen had not failed after 105 cycles, when the test was stopped. The fatigue life for this specimen was taken as 105 cycles into calculation of the average fatigue life, together with the fatigue lives of the specimens that failed in the transition. Even though the recorded fatigue lives have limited statistical significance, the comparison of configurations 2a, 2b and 2c with respect to the fatigue lives associated with failure at and away from the core junction indicates that there is a correlation between localised failure at core junctions and a reduced fatigue life. 8. Discussion and conclusion The influence of local effects on the failure behaviour of sandwich beams with core junctions has been studied experimentally for tensile in-plane loading under both quasi-static and fatigue loading conditions. An attempt was made to predict the failure loads and failure modes based on results from both linear and nonlinear finite element modelling using a point stress criterion originally proposed for the fracture of brittle foam materials. Additionally, simplistic ultimate stress and strain failure criteria were used for the failure assessment. In the quasi-static tests, the failure modes were dependent of the particular material configuration studied, whereas in the fatigue tests visible failure was always confined to the face sheets. Depending on the sandwich configuration, premature failure accumulating at the core junction was observed in the experiments for quasi-static as well as fatigue loading conditions. For a part of the tested configurations, the failure mode shifted from core failure in

Table 7 Fatigue test results

1a 2a

2b 2c a b

Fatigue load Fdyn (N)

Fdyn/Fstat,max

Fatigue life ± SD

Failure mode

Failure location

2700 7100 7100 8900 11590 7100 7100 7100

0.6 0.4 0.4 0.5 0.65 0.6 0.6 (0.4)a

27877 ± 3691 17282 ± 1026 19687 7900 593 4654 10558 ± 10945 65380 ± 37725b

Face Face Face Face Face Face Face Face

Arbitrary (5 of 5/5) Junction (3 of 6/8) Arbitrary (1 of 6/8) Junction (1 of 6/8) Junction (1 of 6/8) Junction (1 of 3/5) Arbitrary (2 of 3/5)

Refers to the static failure load of configuration 2a. cf. text.

Fig. 15. Pictures of failure in the fatigue tests.

failure failure failure failure failure failure failure failure

M. Johannes et al. / Composites Science and Technology 69 (2009) 1447–1457

the quasi-static tests to face failure in the fatigue tests. In the fatigue tests, failure at the core junction was accompanied by a reduced fatigue life. The accuracy of the models used for failure prediction in the quasi-static case was found to depend strongly on the particular sandwich configuration. For specimens using materials with linear (brittle) material behaviour the failure modes and the corresponding failure loads predicted by the linear model agreed reasonably well with the experimental observations. For the cases with distinctly nonlinear material behaviour the predictions based on the linear elastic FE model did not correlate well with the observations. The nonlinear FE model was able to give better results in that case, but various uncertainties remain with respect to the nonlinear FE modelling. In particular the multiaxial stress/strain state at the core junction and the plasticity models used for the foam core materials warrant further examination. Acknowledgements The work presented was supported by: – The Innovation Consortium ‘‘Integrated Design and Processing of Lightweight Composite and Sandwich Structures” (abbreviated ‘‘KOMPOSAND”) funded by the Danish Ministry of Science, Technology and Innovation and the industrial partners Composhield A/S, DIAB ApS (DIAB Group), Fiberline Composites A/S, LM Glasfiber A/S and Vestas Wind Systems A/S. – US Navy, Office of Naval Research (ONR), Grant/Award No. N000140710227: ‘‘Influence of Local Effects in Sandwich Structures under General Loading Conditions & Ballistic Impact on Advanced Composite and Sandwich Structures”. The ONR program manager was Dr. Yapa D.S. Rajapakse. – The polymer foam core materials used in the experimental work were provided by DIAB AB and Röhm GmbH & Co. KG. The support received is gratefully acknowledged. The support received is gratefully acknowledged.

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