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Static indentation and unloading response of sandwich beams
Composites: Part B 35 (2004) 511–522 www.elsevier.com/locate/compositesb
Static indentation and unloading response of sandwich beams Dan Zenkert, Andrey Shipsha*, Karl Persson Division of Lightweight structures, Department of Aeronautical and Vehicle Engineering, Kungl Tekniska Ho¨gskolan, SE-100 44 Stockholm, Sweden Received 31 July 2003; accepted 3 September 2003 Available online 15 April 2004
Abstract This paper deals with analysis of foam core sandwich beams subject to static indentation and subsequent unloading (removal of load). Sandwich beams are assumed continuously supported by a rigid platen to eliminate global bending. An analytical model is presented assuming an elastic-perfectly plastic compressive behaviour of the foam core. An elastic part of indentation response is described using the Winkler foundation model. Upon removal of the load, an elastic unloading response of the foam core is assumed. Also, finite element (FE) analysis of static indentation and unloading of sandwich beams is performed using the FE code ABAQUS. The foam core is modelled using the crushable foam material model. To obtain input data for the analytical model and to calibrate the crushable foam model in FE analysis, the response of the foam core is experimentally characterized in uniaxial compression, up to densification, with subsequent unloading and tension until tensile fracture. Both models can predict load– displacement response of sandwich beams under static indentation and a residual dent magnitude in the face sheet after unloading along with residual strain levels in the foam core at the unloaded equilibrium state. The analytical and FE analyses are experimentally verified through static indentation tests of composite sandwich beams with two different foam cores. The load– displacement response, size of a crushed core zone and the depth of a residual dent are measured in the testing. A digital speckle photography technique is also used in the indentation tests in order to measure the strain levels in the crushed core zone. The experimental results are in good agreement with the analytical and FE analyses. q 2004 Elsevier Ltd. All rights reserved. Keywords: A. Foams; Crushing; C. Analytical modelling; C. Finite element analysis
1. Introduction Owing to low bending stiffness of thin face sheets and low strength of a lightweight core material, sandwich structures are prone to damage when subject to localised loading. Much research effort has been given to this problem in order to model a response of sandwich structures to local loads. An excellent review article by Abrate [1] provides a thorough overview of research work on this subject. The response of sandwich structures to local indentation is mainly dominated by the deformation of the core. Thus, it is important to account for the core deformation in order to obtain accurate prediction of the response. Thomsen [2,3] has recently studied local indentation in both sandwich beams (2D) and sandwich plates (3D). He used a twoparameter Winkler foundation model, which included both transverse and shear stresses. However, only linear elastic response of sandwich beams and panels has been assumed in * Corresponding author. Fax: þ 46-8-20-78-65. E-mail address: [email protected] (A. Shipsha). 1359-8368/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2003.09.006
these analyses. The developed theory has also been successfully verified through experiments. A number of papers have been devoted to using a higher order theory to include the effect of local deformation due to supports and local loads [4,5]. In Ref. [5], this theory was further developed to include a bilinear material behaviour to simulate plasticity. Soden [6] presented an analytical model for indentation of a sandwich beam. He also coupled the model with a failure criterion to obtain closed form solutions for the strength of sandwich beams under concentrated loads. Later, Shuaeib and Soden [7] extended this work and used an linear elastic-perfectly plastic foundation for analysis of static indentation of sandwich beams. Steeves and Fleck [8] used a perfectly plastic foundation in their model to predict failure of sandwich beams in three-point bending. The model includes both the localised deformation due to the load point and the overall bending of the specimen. Some studies have been concerned with the analysis of indentation of sandwich structures by means of finite element (FE) methods. Mines and Alias [9] studied a local indentation of
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sandwich beams under three-point bending using the FE code ABAQUS. The effect of progressive crushing of the core has been implemented through the FOAM material model. In the present paper, a linear elastic-perfectly plastic model is adopted for modelling of static indentation and for predicting the load – displacement response of sandwich beams. When the core undergoes inelastic deformation (core crushing) during indentation, there is a residual dent in the face sheet usually observed after unloading. This study is aimed to extend the indentation model in order to predict the unloading response along with a residual dent magnitude, an extent of the crushed core zone and residual strain levels in the foam core at the unloaded equilibrium state.
2. Experimental study 2.1. Materials and test specimens Two sandwich configurations with different types of foam core were considered in the experimental study. As a core material, Rohacell WF51 PMI foam (Ro¨hm GmbH) and Divinycell H60 PVC foam (DIAB) were used. Both foams have rigid closed-cell structure and are typically used for aerospace and marine applications, respectively. Both sandwich configurations were fabricated with quasi-isotropic GFRP face sheets comprised from Devold DBLT-800 non-crimp fabrics impregnated with Dion 9500 vinylester. Sandwich panels were manufactured by means of vacuum infusion. Beam specimens were cut out of the panels using a diamond blade saw. The material properties and specimen geometry are presented in Table 1. The bending stiffness (flexural modulus) of the face sheets Df was obtained from threepoint bending tests of face laminates conducted according to
Table 1 The mechanical properties of sandwich constituents WF51
H60
Description
Ec (MPa)
75
60
Ert (MPa)
0.40
0.63
Erc (MPa)
16
5
sp (MPa) 1R (– ) 1d ( –) r (kg/m3) tf (mm) b (mm) tc (mm) Df (N mm)
0.90 0.60 0.7 52 2.3 48.7 50 1.60 £ 104
0.75 0.26 0.5 60 2.4 46.2 50 2.22 £ 104
Young’s modulus of virgin foam in compression Young’s modulus of crushed foam in tension Young’s modulus of crushed foam in compression Plateau stress in compression Residual strain Densification strain Density Face sheet thickness Width of beam Core thickness Face sheet bending stiffness
guidelines given in ASTM D790-92 standard [10]. The standard requires that the span length is at least 32 times greater than the face thickness. Two span lengths of 90 and 120 mm were used in the testing. 2.2. Characterisation of foam core 2.2.1. Elastic-perfectly plastic response The analytical model presented in this paper is based on an assumption that the foam core under compressive loading can be characterised by an elastic-perfectly plastic deformation model. Thus, in uniaxial compression test of the foam, the stress – strain curve has three distinct regimes as shown in Fig. 1(a). The foam undergoes a linear elastic deformation up to an elastic strain limit 1p (regime 1) that is usually in the order of 2 – 3%. After the linear elastic strain limit is exceeded, the crushing of the foam core is initiated at almost constant stress, the plateau stress sp ; regime 2. At very high compressive strains, usually around 50– 75%, the crushed core cells start to come in contact and are being compacted, regime 3. This strain level, denoted the densification strain 1d ; is associated with a rapid increase in stiffness. If the load is removed at the end of crushing plateau, the unloading curve for the foam will follow the path 4 as shown in Fig. 1(a). After a complete unloading to zero stress level, there is a residual strain 1R in the foam. If tensile stresses are then applied to crushed foam, the tensile stress – strain response is different than a tensile response of uncrushed foams and depends on the type of foam material. For some foams, the tensile response after crushing is similar to an elastic-perfectly plastic material, and for others, more like a linear elastic type. In general, though, the modulus of the crushed foam is significantly lower than the one for uncrushed foam material [11,12]. 2.2.2. Progressive crushing mechanism Some foam materials exhibit a tendency to plastically deform (crush) in a progressive manner. This implies that crushing of foam is initiated in one particular cell layer, usually the most ‘weak’ layer rather than in a whole volume of the foam. This cell layer is crushing at almost constant stress sp whereas the local strain in that layer rapidly increases from 1p to 1d : When this layer is fully crushed, the opposite cell walls get in contact causing an increase in stiffness and stress which triggers crushing of the next cell layer. In essence, deformations in one single layer undergo the entire regime 2, Fig. 1(a), while all other layers are still in elastic strain state 1p : During indentation or impact of sandwich structures, the stress and strain fields in the core are non-uniform; the largest stress and strain magnitude is observed underneath the load point. Thus, crushing of the core is initiated at this location and then grows around the load point [11]. An important feature of the progressive crushing phenomenon is the presence of a distinct boundary between crushed (densification strain state) and uncrushed
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Fig. 1. (a) Schematic compressive stress–strain behaviour of foams and (b) progressive layer-by-layer crushing during increased compressive strain.
(elastic strain state) core. This feature is exploited in the analytical model presented in the paper. 2.2.3. Experimental characterisation of foam cores Mechanical properties of H60 and WF51 foams were characterised through uniaxial compression tests. Cylindrical specimens were cut from the same core material blocks as used for the sandwich beam specimens. The cylinders were carefully machined in a lathe in order to achieve the geometry shown in Fig. 2. A waist in the specimens was produced using a cylindrical tool with a radius of 12 mm. Aluminium cylinders, bonded to the foam specimens as shown in Fig. 2, were used for fixing the specimens in a testing machine. The specimens were loaded in uniaxial compression until the foam material in the waist part of the specimen was fully crushed and compressed up to the densification strain. The compressive stress – strain response was sampled during the tests. The compression of the specimen was followed by unloading to the zero stress state with subsequent loading in tension or again in compression. Thus, the stiffness characteristics of the crushed foam material Ert (in tension) and Erc (in compression) were measured and given in Table 1. Fig. 3 shows a schematic stress – strain curve from a test and a typical response from a compression test of Rohacell WF51 foam. More detailed description of the testing of crushed foam materials is provided in Ref. [11].
a prescribed displacement magnitude (maximum indentation) was reached, the beams were unloaded at a rate of 20 mm/min. After the unloading, a residual dent in the face sheet was clearly observed. The residual dent magnitude was measured using two different techniques: (1) moving a dial gauge over the residual dent surface and (2) associating a maximum dent magnitude with the displacement from the testing machine that corresponds to the zero loads during the unloading of specimens. In the latter technique, an instantaneous magnitude of the residual dent after unloading is obtained, so the relaxation and creep effects in the foam can be neglected. It was observed that measuring the residual dent magnitude after 5– 10 min after unloading gives 25 – 40% lower value if compared with the instantaneous value. Thus, a correct model for predicting the residual dent magnitude should account for relaxation and creep in the crushed foam. 2.4. Digital speckle photography analysis A digital speckle photography (DSP) analysis was used in static indentation tests for measuring compressive strains in the foam core and crushed core zone. The DSP measurements were conducted by means of the DSP system ARAMIS (GOM mbH, Germany). The DSP analysis relies
2.3. Static indentation tests The indentation tests of sandwich beams were performed in an INSTRON universal-testing machine. The test set-up is schematically shown in Fig. 4. The sandwich beam was placed on a rigid steel block providing a continuous support and eliminating an overall bending of a beam. The beams were loaded through a steel cylinder with 25 mm diameter. A constant crosshead displacement rate of 2 mm/min was used in all tests. The indentation tests were conducted in a displacement control. When
Fig. 2. Uniaxial compression test of foam specimens.
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Fig. 3. (a) A schematic stress–strain response from the compression–tension test and (b) typical stress–strain curve from a compression test of Rohacell WF51 foam.
on comparison of pairs of digital images taken by a charged coupled device camera during the deformation progress. In order to enhance the image analysis, a random speckle pattern was applied using a spray paint on the foam core surface, underneath the indentation area, prior to testing. The initial image captured before deformation was divided into domain of sub-images. Using a digital image correlation algorithm, the corresponding position of each sub-image within the domain in the deformed image could be found providing the in-plane relative displacement field. The in-plane strain components 1ii were derived by differentiating the displacement vector field. The resolution of displacement measurements using the DSP analysis was about 1/15,000 of the total image size (approx. 60 £ 40 mm2). This gave the accuracy in measured elastic strain magnitudes of about ^ 0.1– 0.3% (absolute values of strain in %). For a crushed zone with higher strains, the strain field was quite noisy and therefore filtered, which reduced the accuracy of measured strain magnitudes to ^ 0.5– 1.0%.
at the lower boundary of the core are restricted (simulating a continuous support by a rigid foundation), and no overall bending of a sandwich beam is considered. The core is assumed to remain perfectly bonded to the face sheet during indentation and unloading. Deflection of the face sheet with bending stiffness Df is assumed linear elastic. Also, the face sheet is considered to remain undamaged. A compressive response of the foam core is elastic-perfectly plastic and the progressive core crushing mechanism is assumed. The loading – unloading sequence is modelled through the following three stages. A prescribed load or displacement is applied in the first stage simulating the indentation load. The second stage is a hypothetical state, where the load is removed and the face and core are detached. Assuming elastic bending, the face sheet deflects back and retains its original position, thus leaving an open gap between the face sheet and the foam core. In the third stage, the face and the core are ‘bonded’ together in order to find the equilibrium state and resolve the residual dent and strains in the foam core. The stages 2 and 3 are treated as purely elastic.
3. Analytical model: approach and assumptions
3.1. Stage 1—indentation loading
The analytical model considers bending of a transversely isotropic face sheet bonded to a foam core and subject to a concentrated line-load as shown in Fig. 5. All displacements
Here, a simple indentation model is employed, similar to the model described in Abrate [1] or Shuaeib and Soden [7]. The model assumes an elastic Winkler foundation for
Fig. 4. Test set-up for sandwich beam indentation.
Fig. 5. A schematic of the indentation model.
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the elastic core, and a perfectly plastic foundation for the part of the core that undergoes crushing, as schematically shown in Fig. 5. 3.1.1. Elastic solution When the load P is small, the entire foundation is elastic. The governing equation for a beam on an elastic foundation reads Df
d4 we1 þ kwe1 ¼ 0 dx4e
ð1Þ
where we1 is the face sheet deflection (the index 1 refers to stage 1) and k is the foundation modulus. In the case considered in this paper, k may roughly be estimated as k¼
ð2Þ
where Ec is the core elastic modulus and tc is the core thickness. The solution of Eq. (1) is we1 ðxe Þ ¼ e2lxe ðA sin lxe þ B cos lxe Þ þ elxe ðC sin lxe þ D cos lxe Þ
ð3Þ
Pp ¼
2Ec 1p l
ð6Þ
3.1.2. Perfectly plastic solution At the increased load, a part of the core with length 2a undergoes plastic deformation (crushing). The crushing core exerts a uniform constant reaction sp on the face sheet on the length 2a (a perfectly plastic foundation). The governing equation for the face sheet bending on a perfectly plastic foundation reads d4 wp1 þ sp ¼ 0 dx4p
ð7Þ
where sp is the plateau stress or compressive yield stress of the core material. The general solution to Eq. (7) is wp1 ðxp Þ ¼ 2
sp x4p x3p x2p þ C1 þ C2 þ C3 xp þ C4 24 3 2
The boundary conditions are now
with
l4 ¼
prior to plastic deformation can be determined as
Df
Ec tc
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wp1 ð0Þ ¼ a;
k 4Df
The constants A; B; C and D in Eq. (3) are determined from boundary conditions which require the solution remains bounded at infinity we1 ðxe ! 1Þ ¼ 0;
w0e1 ðxe
! 1Þ ¼ 0
and that the slope at the origin be zero because of symmetry w0e1 ðxe ¼ 0Þ ¼ 0: Further, the deflection of the face sheet in the origin is
w0p1 ð0Þ
wp1 ðxp Þ ¼ a 2
sp x4p Px3p x2p þ þ C2 24 12Df 2
which gives that A ¼ B ¼ a: Thus, the elastic solution can be written as
w0p1 ðaÞ ¼ w0e1 ð0Þ
It can be seen that the elastic solution is a damped harmonic function which means that the displacement function exhibits an ‘overshoot’ at some co-ordinate xe so that there will be tensile normal stresses in the core. The contact force required to produce the initial deflection a is twice the magnitude of the shear force in the origin and can be determined as ! P d3 we1 ¼ Df !P ¼ 8Df l3 a ð5Þ 2 dx3e xe ¼0 The onset of plastic deformation (core crushing) occurs at the condition we1 ð0Þ ¼ 1p tc : By combining Eq. (2), second part of Eqs. (3) and (5), the maximum contact force
¼ xp ¼0
P 2Df
ð9Þ
A general equation for the face deflection is obtained by combining the elastic and plastic solutions of Eqs. (4) and (9). In these two expressions, there is a total of five unknowns (a; A B; C2 and P). The following four continuity equations can be used to solve for four of them wp1 ðaÞ ¼ we1 ð0Þ
ð4Þ
!
so that
we1 ðxe ¼ 0Þ ¼ a
we1 ðxe Þ ¼ a e2lxe ðsin lxe þ cos lxe Þ
¼ 0; and
d3 wp1 dx3p
ð8Þ
w00p1 ðaÞ ¼ w00e1 ð0Þ 000 w000 p1 ðaÞ ¼ we1 ð0Þ
A fifth condition is obtained by requiring that at the point of transition between elastic and plastic core, at x ¼ a; the transverse normal strain be equal to the maximum strain before crushing wp1 ðaÞ ¼ we1 ð0Þ ¼ 1p tc so that B ¼ 1p t c : Using these conditions, Eqs. (4) and (9) can be solved simultaneously for indentation force P for a given length of the plastic zone a: Also, the plastic zone a can be calculated for a given load P:
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3.1.3. Thickness of elastic and plastic core Assuming the progressive crushing mechanism of the foam, the crushed core is all in a state corresponding to the densification strain 1d ; while the core underneath the crushed zone is strained to the elastic limit 1p : In the elastic zone, x . a; the core is in an elastic state with a strain 1p at xe ¼ 0: For x ¼ a; the deflection can be written as w1 ðaÞ ¼ 1p tc
ð10Þ
The following simple approach is used to calculate the thickness of the crushed core layer for a given face sheet deformation w1 as shown in Fig. 6. It is obvious that the thickness of the core in elastic and plastic states is a function of the displacement w1 ðxÞ: They can be related through strains in the different regions as tp1 ðxÞ ¼
ðw1 ðxÞ 2 tc 1p Þð1 2 1d Þ ð1d 2 1p Þ
ð11Þ
Since w1 is a function of x; both tp1 and te1 are also functions of x: 3.2. Stage 2—removal load and detachment of face sheet from core In this stage, the load is removed and the face sheet is hypothetically detached from the core, as schematically shown in Fig. 7. After detaching the face sheet, the elastic part of the core relaxes retaining a zero strain level. The crushed core on the other hand, will remain at the residual strain level 1R ; as shown in Fig. 1. Since the face sheet was assumed to bend elastically, it will bounce back to a zero displacement level after detaching from the core, thus leaving a gap between the face and the core. At this stage, the elastic core thickness can be obtained as te2 ¼
te1 1 2 1p
ð12Þ
Fig. 7. A schematic of deformations and strains in a sandwich beam at the stage 2.
be calculated knowing the residual strains as tp2 ¼
tp1 ð1 2 1R Þ 1 2 1d
ð13Þ
3.3. Stage 3—attachment of face to core In this stage, the stage 2 is used as the base configuration and all displacements and strains are referred to the stage 2. In the third stage, an equilibrium state for which the face sheet and the core share the same displacement w3 is sought. In other words, the face sheet is given a displacement w3 ðxÞ and the core is ‘pulled up’ to the same position as schematically shown in Fig. 8. The function w3 ðxÞ for a state of equilibrium was found using an approximate method. From measurements of the residual dent after indentation or impact, it was observed that the dent has a certain length L; and a certain maximum amplitude b for x ¼ 0: The measurements also implied that the shape of the dent can be expressed approximately by a third order polynomial. Assuming that the function w3 ðxÞ can be expressed as w3 ðxÞ ¼ Ax3 þ Bx2 þ Cx þ D with the boundary conditions w3 ð0Þ ¼ b w03 ð0Þ ¼ 0 w3 ðLÞ ¼ 0 w03 ðLÞ ¼ 0
At the transition point between the plastic and elastic core, at x ¼ a; the thickness of the elastic core in this stage is te2 ¼ tc : The thickness of the crushed core in this stage can
yielding that
Fig. 6. A schematic of deformations and strains in a sandwich beam at the stage 1.
Fig. 8. A schematic of deformations and strains in a sandwich beam at the stage 3.
w3 ðxÞ ¼
2 bx 3 3bx 2 2 þ b; L3 L2
x$0
ð14Þ
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The unknown residual dent magnitude b and length L can be found using the Eq. (14) and applying the principle of minimum total energy.
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in compression. The displacement d can thus be written as
d ¼ te3 þ tp3 2 te2 2 tp2 or substituting Eq. (17), as
3.3.1. Face sheet bending strain energy At the third stage, the bending strain energy of the face sheet can be expressed as !2 1 ðL d2 w3 Pb ¼ D dx ð15Þ 2 0 f dx2 The elastic part of the core, for a , x , L; is 1 ð L ð tc 1 ðL ðtc w3 w3 Pe ¼ s1 dx ¼ E dx 2 a 0 2 a 0 c tc tc ¼
1 ðL kðw3 Þ2 dx 2 a
Since this a Winkler type foundation model, the same stress is applied to both the elastic and the plastic part of the core. We can then write s s d ¼ te2 1e3 þ tp2 1p3 ¼ te2 þ tp2 ð19Þ Ec ER The equivalent foundation modulus kR is defined by tp2 21 s t Ec ER kR ¼ ¼ e2 þ ¼ ð20Þ d Ec ER te2 ER þ tp2 Ec
ð16Þ
The stored strain energy in the elastic/plastic part of the core, x , a; remains. The elastic part of the core with thickness te2 and zero strain at the stage 2 now changes to a thickness te3 ; while the plastic core with the thickness tp2 that changes to tp3 : The following relation for compatibility can be written using the schematic in Fig. 9. te3 þ tp3 þ w3 ¼ te2 þ tp2 þ d þ w3 ¼ tc where d is defined as the displacement of the core between stages 2 and 3. The thickness of the core layers in the elastic and plastic parts of the core is now expressed as te3 ¼ te2 ð1 2 1e3 Þ and tp3 ¼ tp2 ð1 2 1p3 Þ
d ¼ te2 1e3 þ tp2 1p3
ð17Þ
If the strain 1p3 is positive then ER ¼ ERt ; otherwise ER ¼ ERc : The point of transition, i.e. for 1p3 ¼ 0; is found by numerical iteration and is denoted xg : The total stored elastic energy in the core is then found by integration of 1 ða Pe ¼ k ðdÞ2 dx ð21Þ 2 0 R The total stored energy in the system will then be !2 1 ðL d2 w 3 1 ðL P¼ Df dx þ kðw3 Þ2 dx 2 2 0 2 a dx 1 ða þ k ðdÞ2 dx ð22Þ 2 0 R Transverse equilibrium strains are found from Eq. (19)
ERt
and tp3 t 1e3 ¼ 1 2 e3 and 1p3 ¼ 1 2 te2 tp2
1p3 ¼
ð18Þ
The modulus of the elastic layer of the core remains equal to Ec while the plastic part will have a modulus equal to ERt providing the plastic core part will be in tension and with modulus ERc if the plastic part of the core will be
d tp2 te2 þ Ec ERt
; 1e3 ¼
Ec
d tp2 te2 þ Ec ERt
; ð23Þ
0 , x , xg
1p3 ¼ ERc
d d tp2 ; 1e3 ¼ tp2 ; te2 te2 þ Ec þ Ec ERc Ec ERc
ð24Þ
xg , x , a Energy methods based on assumed displacement functions, such as that used in the analysis of unloading, provide powerful tools for solving complex problems but do not in general lead to exact solutions as the conditions of equilibrium are only approximately satisfied. 3.4. Finite element modelling
Fig. 9. Geometry used to find strains in the elastic–plastic part of the core at stage 3.
The developed analytical solutions were verified by the FE analysis of the static indentation of sandwich beams. The FE package ABAQUS was used in this study. The FE analysis was carried out for the sandwich configurations as given in Table 1. Plane strain condition was assumed in the FE analysis. Four-node bilinear plane strain elements
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(CPE4) were used. The two-dimensional mesh consisted of 1200 elements, with two elements through the thickness of each face, 20 elements through the thickness of the core, and 50 elements along the length of the sandwich beam. The FE mesh was refined towards the indentation point. This mesh was selected on the basis of a convergence study with respect to the number of elements, and the bias ratio. All degrees of freedom were constrained at the lower boundary of the model, simulating a rigid fixed foundation. Due to the symmetry conditions, only half of the sandwich beam was modelled. The indenter was modelled as a rigid cylinder with diameter of 25 mm. All degrees of freedom of the indenter were constrained, except for the translation in vertical direction (normal to the upper face sheet plane). The interaction between the indenter and the upper face sheet was modelled using a contact algorithm implemented in ABAQUS. The FE model is shown in Fig. 10. The plastic response of the core material was modelled using *CRUSHABLE FOAM and *CRUSHABLE FOAM HARDENING options in ABAQUS, version 6.3. This material model was calibrated using the test data from a uniaxial compression of the foam material. The data was given in terms of uniaxial compression yield stress versus corresponding logarithmic plastic strain. The face sheets were modelled as linear elastic. Since the sandwich structure undergoes large deformations, geometrically non-linear analysis was conducted using the *NLGEOM option. The unloading after indentation was simulated applying a reversed prescribed displacement of the rigid cylinder. The contact interaction between the face and the cylinder was computed automatically by ABAQUS. The stiffness characteristics of the volume of the core that was crushed at the loading step were kept unchanged. Thus, the FE model was expected to provide a ‘more stiff’ response for the unloading curve than observed in the experiments. Subsequently, the residual dent magnitude will be somewhat overestimated.
4. Comparison of theoretical and experimental results 4.1. Elastic response to indentation load The critical load Pcr at which core crushing first occurs was measured from the indentation tests. It was taken as a maximum load observed for the linear load – deflection response. Theoretical predictions of the critical load were obtained using Eq. (6). A comparison of experimentally measured and predicted critical loads is presented in Table 2. Theoretical and FE results are very dependent on both the correct yield strength of the core sp and on the elastic properties of the face sheet. In general, the predicted values are slightly underestimated but still show good agreement with experimental results. 4.2. Elastic– plastic response to indentation load For higher loads ð. Pcr Þ; the load –deflection response becomes non-linear owing to the core crushing. Typical load –deflection curves for two tested sandwich configurations are shown in Fig. 11. The maximum indentation (deflection magnitude) in the tests was 7 mm. At higher deflections, the onset of delaminations and fibre breakage was observed in the face sheet causing a noticeable reduction in the stiffness. The load – deflection response from the tests exhibits a ‘load dip’ at the onset of core crushing. A similar behaviour was observed from the uniaxial tests of foam specimens which show that the initiation stress (stress level at which first cell layer crushes) is slightly higher than the plateau stress, Fig. 3(b). The analytical and FE models do not account for this effect exhibiting a smooth transition from linear elastic to non-elastic response of sandwich beams under indentation loading. However, the predicted load – deflection response is in very good agreement with experimental data, Fig. 11. The size of the plastic zone (crushed core) at different load levels was measured from the indentation tests using the DSP equipment. These measurements are plotted against the indentation load in Fig. 12 along with predictions from the analytical model and FE analysis. The experimentally measured size of the crushed core zone is in good agreement with analytical predictions. The results from the FE study slightly underestimate the size of the crushed core zone. Table 2 Critical load at which core crushing was first observed and comparison of the experimental data with analytical and finite element predictions
Fig. 10. Finite element model of static indentation test.
Configuration
Pcr (experiment) (kN)
Pcr (analytical) (kN)
Pcr (FE) (kN)
Rohacell WF51 core Divinycell H60 core
1.60 1.45
1.40 1.14
1.35 1.10
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Fig. 11. Static indentation response of the sandwich beams with (a) Rohacell WF51 foam core and (b) Divinycell H60 foam core.
4.3. Residual dent Experimental measurements have shown that the amplitude of the residual dent decreases with time owing to creep or relaxation effects in the crushed foam core. Therefore, the results of the residual dent measurements are very sensitive to the time passed after the indentation test. In this study, the measurements of the residual dent were performed directly in the test machine and immediately after unloading to the zero load level, thus, measuring the instantaneous residual dent magnitude. These values were used for comparison with predictions from analytical and FE models which do not account for the effect of creep. It should be mentioned that a steady-state magnitude of the residual dent is significantly lower than the instantaneous dent by as much as 25 – 40%. Fig. 13 shows the magnitude of the residual dent measured in the tests at different maximum indentation magnitude.
The correlation between experimentally measured dent magnitudes and the analytically predicted values is rather good. In the H60 case, the analytical model underestimates the dent magnitude somewhat, whereas the agreement for the WF51 configuration is much better. As it was anticipated, the FE predictions considerably overestimate the residual dent magnitude due to very simplified and rough approach used for modelling of the unloading response. The length of the residual dent was also measured in experiments after indentation at different magnitude and compared with predictions from the analytical model, Fig. 14. The agreement between measured and predicted values was good. The FE predictions (not presented in the plot) overestimate the length of the residual dent due to aforementioned reasons. In general, the predictions of residual dent parameters (magnitude and length) are very encouraging, especially considering that the analytical approach is quite simple and based on many assumptions.
Fig. 12. Indentation load vs. half-length of plastic zone predicted and measured for (a) WF51 and (b) H60 sandwich configurations.
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Fig. 13. Magnitude of residual dent plotted against the indentation magnitude for (a) WF51 and (b) H60 specimen configurations.
4.4. Residual strains in the crushed core The DSP analysis was used for experimental evaluation of residual strains in the crushed core of sandwich beams after indentation and unloading. Fig. 15 shows the residual strains distribution in the crushed core of WF51/GFRP beam immediately after unloading and after relaxation during 10 min. The plots are presented in un-deformed co-ordinates; therefore, the residual dent is not visible. It can be observed that the residual strains along the crushed core zone are quite uniform, reducing towards the exterior of the zone. Two bands of somewhat more compressed core are visible in the plots. The analytical model presented in this study was used to obtain an estimate of residual strains in the crushed core. The residual strains were estimated using the following expression 1residual ¼ 1 2
tp3 ð1 2 1d Þ tp1
ð25Þ
where tp3 and tp1 are the thickness of the crushed core at stage 3 and stage 1, respectively. This simple estimate provided quite good agreement for residual strains measured in the middle, directly under the indentation point. Thus, the maximum magnitude of residual strains in WF51/GFRP beams was measured to be about 27% immediately after unloading. The predicted values were in the order of 30%. For H60/GFRP beams, the agreement was better; the measured and predicted values of residual strains in the middle were in the order of 12 –13%. The comparison for the residual strains in the crushed core is graphically shown in Fig. 16. At this stage, the analytical model is not valid to calculate residual strains along the whole crushed core zone. The model suggests that the residual strains increase towards the exterior of the crushed core zone that is not physical. Inherent features of the presented model may explain this discrepancy. Thus, the thickness of the crushed core at stage 1, tp1 ; is a function of forth polynomial whereas
Fig. 14. Length of residual dent plotted against the indentation magnitude for (a) WF51 and (b) H60 specimen configurations.
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Fig. 15. DSP image of residual strains in the crushed foam core of WF51/GFRP beam: (a) after unloading and (b) after relaxation (10 min). The plots are given in un-deformed co-ordinates.
the thickness of crushed core at stage 3, tp3 ; is a function of third polynomial (assumed function for the residual dent). It should be possible to overcome the problem and use the same degree polynomial function, which can improve the agreement with experimental data, however, for now it is left beyond this study.
5. Discussion and conclusions An analytical/numerical approach to modelling the indentation and unloading of sandwich beams is presented in this paper and validated by experimental results. The presented analytical model is based on an elastic-perfectly plastic Winkler type foundation and utilises the inherent progressive crushing mechanism of foams. There are several simplifications used in the model and also several facts that worth noticing that have an effect on the validity of the modelling approach. The Winkler foundation assumption contains a few rather rough approximations. First of all, transverse shear stresses are not accounted for. Secondly, the foundation modulus k (Eq. (2)) is due to its definition geometry dependent, and various definitions of the foundation modulus have been used over the years by different researchers in the pursuit of models for sandwich behaviour, e.g. local loads and wrinkling. In here, a very simple definition has been used. Thirdly, lateral deformations are considered negligible thus Poisson’s ratio is taken to be zero. The model is only valid for foam cores that exhibit constant crushing stress (perfectly plastic materials). Other material behaviour would require implementation of a hardening rule and it is left beyond the scope of this study. When modelling the progressive crushing of the core, it is assumed that the core either comprises of regions with densification strain or elastic limit strain, i.e. 1d or 1p : In reality, the strain levels will not assume discrete values,
but rather have intermediate zones of varying strain magnitudes. The results obtained herein imply that the simplified approach can be sufficient to describe the crushing mechanism of foam cores. Mechanical behaviour of foam materials is much more complex than for solid materials. Thus, for analysis presented in this study, extensive knowledge about mechanical properties and behaviour of a foam material is required. Many parameters can only be extracted by quite complex experimental procedures. The outcome of the simulation depends highly on these parameters, so great effort was made to establish, for example, Young’s modulus for compressed foam, etc. In some tests, strains of approximately 70% were measured based on displacement of the crosshead and specimen geometry. Evidently, such measurements are hard to conduct without introducing some degree of error. Although these facts contribute to the overall output error, the results obtained for two different
Fig. 16. Distribution of residual strains along the crushed core zone from the DSP analysis.
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material combinations, seem to justify the use of a Winkler type foundation. To find the equilibrium shape of the deflected face sheet, energy methods were utilised. Strain/bending energies are stipulated and the total energy of the system is minimised with respect to unknown variables b and L: Energy methods provide powerful tools for solving complex problems but do not in general lead to exact solutions as the conditions of equilibrium are only approximately satisfied. Upon unloading of indented sandwich beams, an extensive creep has been observed in experiments. The effect of creep has not been implemented in the model due to lack of material data. It must be emphasised that implementing creep as a mode of deformation is vital in future work. During the experimental phase of this study, two different methods of measuring the residual dent were used. One of the methods presented in the paper allows for measuring almost instantaneous value of the dent magnitude after unloading.
Acknowledgements The authors (Dan Zenkert and Andrey Shipsha) are grateful for financial support from the ONR Research Program on ‘Composites for Marine Structures’ through the program manager Dr Y.D.S. Rajapakse. Dr Victor Rizov (Univ. of Architecture, Civil Engineering and Geodesy, Bulgaria) is acknowledged for help with finite element analysis.
References [1] Abrate S. Localized impact on sandwich structures with laminated facings. Appl Mech Rev 1997;50(2):69 –82. [2] Thomsen OT. Analysis of local bending effects in sandwich panels subjected to local concentrated loads. In: Weissman-Berman D, Olsson K-A, editors. Proceedings of the Second International Conference on Sandwich Construction. UK: EMAS Ltd 1992. p. 417–40. [3] Thomsen OT. Theoretical and experimental investigation of local bending effects in sandwich plates. Compos Struct 1995;30:85–101. [4] Frostig Y, Barush M, Vilnay O, Sheinman I. Bending of nonsymmetric sandwich beams with transversely flexible core. ASCE J Engng Mech 1991;117(9):1931–52. [5] Schwarts-Givli H, Frostig Y. Behaviour of unidirectional sandwich panels with a bi-linear soft core—higher order approach. ASME ICEME’2000 Conference in Orlando, November 5 – 10, ASME AERO/AMD AD-vol. 62/AMD vol. 245; 2000. p. 263–76. [6] Soden PD. Indentation of composite sandwich beams. J Strain Anal 1996;31(5):353– 60. [7] Shuaeib FM, Soden PD. Indentation failure of composite sandwich beams. Compos Sci Technol 1997;57:1249–59. [8] Steeves CA, Fleck NA. Failure modes in sandwich beams with composite face-sheets and PVC foam cores. ASME ICEME’2000 Conference in Orlando, November 5–10, ASME AERO/AMD ADvol. 62/AMD vol. 245; 2000. p. 21–8. [9] Mines RAW, Alias A. Numerical simulation of the progressive collapse of polymer composite sandwich beams under static loading. Composites, Part A 2002;33:11– 26. [10] Annual book of ASTM standards. Technical standard ASTM D790-92. [11] Shipsha A, Hallstro¨m S, Zenkert D. Failure mechanisms and modelling of impact damage in sandwich beams—a 2D approach. Part I. Experimental investigation. J Sandwich Struct Mater 2003; 5(1):7–32. [12] Shipsha A, Hallstro¨m S, Zenkert D. Failure mechanisms and modelling of impact damage in sandwich beams—a 2D approach. Part II. Analysis and modelling. J Sandwich Struct Mater 2003;5(1): 33 –52.
Static indentation and unloading response of sandwich beams Dan Zenkert, Andrey Shipsha*, Karl Persson Division of Lightweight structures, Department of Aeronautical and Vehicle Engineering, Kungl Tekniska Ho¨gskolan, SE-100 44 Stockholm, Sweden Received 31 July 2003; accepted 3 September 2003 Available online 15 April 2004
Abstract This paper deals with analysis of foam core sandwich beams subject to static indentation and subsequent unloading (removal of load). Sandwich beams are assumed continuously supported by a rigid platen to eliminate global bending. An analytical model is presented assuming an elastic-perfectly plastic compressive behaviour of the foam core. An elastic part of indentation response is described using the Winkler foundation model. Upon removal of the load, an elastic unloading response of the foam core is assumed. Also, finite element (FE) analysis of static indentation and unloading of sandwich beams is performed using the FE code ABAQUS. The foam core is modelled using the crushable foam material model. To obtain input data for the analytical model and to calibrate the crushable foam model in FE analysis, the response of the foam core is experimentally characterized in uniaxial compression, up to densification, with subsequent unloading and tension until tensile fracture. Both models can predict load– displacement response of sandwich beams under static indentation and a residual dent magnitude in the face sheet after unloading along with residual strain levels in the foam core at the unloaded equilibrium state. The analytical and FE analyses are experimentally verified through static indentation tests of composite sandwich beams with two different foam cores. The load– displacement response, size of a crushed core zone and the depth of a residual dent are measured in the testing. A digital speckle photography technique is also used in the indentation tests in order to measure the strain levels in the crushed core zone. The experimental results are in good agreement with the analytical and FE analyses. q 2004 Elsevier Ltd. All rights reserved. Keywords: A. Foams; Crushing; C. Analytical modelling; C. Finite element analysis
1. Introduction Owing to low bending stiffness of thin face sheets and low strength of a lightweight core material, sandwich structures are prone to damage when subject to localised loading. Much research effort has been given to this problem in order to model a response of sandwich structures to local loads. An excellent review article by Abrate [1] provides a thorough overview of research work on this subject. The response of sandwich structures to local indentation is mainly dominated by the deformation of the core. Thus, it is important to account for the core deformation in order to obtain accurate prediction of the response. Thomsen [2,3] has recently studied local indentation in both sandwich beams (2D) and sandwich plates (3D). He used a twoparameter Winkler foundation model, which included both transverse and shear stresses. However, only linear elastic response of sandwich beams and panels has been assumed in * Corresponding author. Fax: þ 46-8-20-78-65. E-mail address: [email protected] (A. Shipsha). 1359-8368/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2003.09.006
these analyses. The developed theory has also been successfully verified through experiments. A number of papers have been devoted to using a higher order theory to include the effect of local deformation due to supports and local loads [4,5]. In Ref. [5], this theory was further developed to include a bilinear material behaviour to simulate plasticity. Soden [6] presented an analytical model for indentation of a sandwich beam. He also coupled the model with a failure criterion to obtain closed form solutions for the strength of sandwich beams under concentrated loads. Later, Shuaeib and Soden [7] extended this work and used an linear elastic-perfectly plastic foundation for analysis of static indentation of sandwich beams. Steeves and Fleck [8] used a perfectly plastic foundation in their model to predict failure of sandwich beams in three-point bending. The model includes both the localised deformation due to the load point and the overall bending of the specimen. Some studies have been concerned with the analysis of indentation of sandwich structures by means of finite element (FE) methods. Mines and Alias [9] studied a local indentation of
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sandwich beams under three-point bending using the FE code ABAQUS. The effect of progressive crushing of the core has been implemented through the FOAM material model. In the present paper, a linear elastic-perfectly plastic model is adopted for modelling of static indentation and for predicting the load – displacement response of sandwich beams. When the core undergoes inelastic deformation (core crushing) during indentation, there is a residual dent in the face sheet usually observed after unloading. This study is aimed to extend the indentation model in order to predict the unloading response along with a residual dent magnitude, an extent of the crushed core zone and residual strain levels in the foam core at the unloaded equilibrium state.
2. Experimental study 2.1. Materials and test specimens Two sandwich configurations with different types of foam core were considered in the experimental study. As a core material, Rohacell WF51 PMI foam (Ro¨hm GmbH) and Divinycell H60 PVC foam (DIAB) were used. Both foams have rigid closed-cell structure and are typically used for aerospace and marine applications, respectively. Both sandwich configurations were fabricated with quasi-isotropic GFRP face sheets comprised from Devold DBLT-800 non-crimp fabrics impregnated with Dion 9500 vinylester. Sandwich panels were manufactured by means of vacuum infusion. Beam specimens were cut out of the panels using a diamond blade saw. The material properties and specimen geometry are presented in Table 1. The bending stiffness (flexural modulus) of the face sheets Df was obtained from threepoint bending tests of face laminates conducted according to
Table 1 The mechanical properties of sandwich constituents WF51
H60
Description
Ec (MPa)
75
60
Ert (MPa)
0.40
0.63
Erc (MPa)
16
5
sp (MPa) 1R (– ) 1d ( –) r (kg/m3) tf (mm) b (mm) tc (mm) Df (N mm)
0.90 0.60 0.7 52 2.3 48.7 50 1.60 £ 104
0.75 0.26 0.5 60 2.4 46.2 50 2.22 £ 104
Young’s modulus of virgin foam in compression Young’s modulus of crushed foam in tension Young’s modulus of crushed foam in compression Plateau stress in compression Residual strain Densification strain Density Face sheet thickness Width of beam Core thickness Face sheet bending stiffness
guidelines given in ASTM D790-92 standard [10]. The standard requires that the span length is at least 32 times greater than the face thickness. Two span lengths of 90 and 120 mm were used in the testing. 2.2. Characterisation of foam core 2.2.1. Elastic-perfectly plastic response The analytical model presented in this paper is based on an assumption that the foam core under compressive loading can be characterised by an elastic-perfectly plastic deformation model. Thus, in uniaxial compression test of the foam, the stress – strain curve has three distinct regimes as shown in Fig. 1(a). The foam undergoes a linear elastic deformation up to an elastic strain limit 1p (regime 1) that is usually in the order of 2 – 3%. After the linear elastic strain limit is exceeded, the crushing of the foam core is initiated at almost constant stress, the plateau stress sp ; regime 2. At very high compressive strains, usually around 50– 75%, the crushed core cells start to come in contact and are being compacted, regime 3. This strain level, denoted the densification strain 1d ; is associated with a rapid increase in stiffness. If the load is removed at the end of crushing plateau, the unloading curve for the foam will follow the path 4 as shown in Fig. 1(a). After a complete unloading to zero stress level, there is a residual strain 1R in the foam. If tensile stresses are then applied to crushed foam, the tensile stress – strain response is different than a tensile response of uncrushed foams and depends on the type of foam material. For some foams, the tensile response after crushing is similar to an elastic-perfectly plastic material, and for others, more like a linear elastic type. In general, though, the modulus of the crushed foam is significantly lower than the one for uncrushed foam material [11,12]. 2.2.2. Progressive crushing mechanism Some foam materials exhibit a tendency to plastically deform (crush) in a progressive manner. This implies that crushing of foam is initiated in one particular cell layer, usually the most ‘weak’ layer rather than in a whole volume of the foam. This cell layer is crushing at almost constant stress sp whereas the local strain in that layer rapidly increases from 1p to 1d : When this layer is fully crushed, the opposite cell walls get in contact causing an increase in stiffness and stress which triggers crushing of the next cell layer. In essence, deformations in one single layer undergo the entire regime 2, Fig. 1(a), while all other layers are still in elastic strain state 1p : During indentation or impact of sandwich structures, the stress and strain fields in the core are non-uniform; the largest stress and strain magnitude is observed underneath the load point. Thus, crushing of the core is initiated at this location and then grows around the load point [11]. An important feature of the progressive crushing phenomenon is the presence of a distinct boundary between crushed (densification strain state) and uncrushed
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Fig. 1. (a) Schematic compressive stress–strain behaviour of foams and (b) progressive layer-by-layer crushing during increased compressive strain.
(elastic strain state) core. This feature is exploited in the analytical model presented in the paper. 2.2.3. Experimental characterisation of foam cores Mechanical properties of H60 and WF51 foams were characterised through uniaxial compression tests. Cylindrical specimens were cut from the same core material blocks as used for the sandwich beam specimens. The cylinders were carefully machined in a lathe in order to achieve the geometry shown in Fig. 2. A waist in the specimens was produced using a cylindrical tool with a radius of 12 mm. Aluminium cylinders, bonded to the foam specimens as shown in Fig. 2, were used for fixing the specimens in a testing machine. The specimens were loaded in uniaxial compression until the foam material in the waist part of the specimen was fully crushed and compressed up to the densification strain. The compressive stress – strain response was sampled during the tests. The compression of the specimen was followed by unloading to the zero stress state with subsequent loading in tension or again in compression. Thus, the stiffness characteristics of the crushed foam material Ert (in tension) and Erc (in compression) were measured and given in Table 1. Fig. 3 shows a schematic stress – strain curve from a test and a typical response from a compression test of Rohacell WF51 foam. More detailed description of the testing of crushed foam materials is provided in Ref. [11].
a prescribed displacement magnitude (maximum indentation) was reached, the beams were unloaded at a rate of 20 mm/min. After the unloading, a residual dent in the face sheet was clearly observed. The residual dent magnitude was measured using two different techniques: (1) moving a dial gauge over the residual dent surface and (2) associating a maximum dent magnitude with the displacement from the testing machine that corresponds to the zero loads during the unloading of specimens. In the latter technique, an instantaneous magnitude of the residual dent after unloading is obtained, so the relaxation and creep effects in the foam can be neglected. It was observed that measuring the residual dent magnitude after 5– 10 min after unloading gives 25 – 40% lower value if compared with the instantaneous value. Thus, a correct model for predicting the residual dent magnitude should account for relaxation and creep in the crushed foam. 2.4. Digital speckle photography analysis A digital speckle photography (DSP) analysis was used in static indentation tests for measuring compressive strains in the foam core and crushed core zone. The DSP measurements were conducted by means of the DSP system ARAMIS (GOM mbH, Germany). The DSP analysis relies
2.3. Static indentation tests The indentation tests of sandwich beams were performed in an INSTRON universal-testing machine. The test set-up is schematically shown in Fig. 4. The sandwich beam was placed on a rigid steel block providing a continuous support and eliminating an overall bending of a beam. The beams were loaded through a steel cylinder with 25 mm diameter. A constant crosshead displacement rate of 2 mm/min was used in all tests. The indentation tests were conducted in a displacement control. When
Fig. 2. Uniaxial compression test of foam specimens.
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Fig. 3. (a) A schematic stress–strain response from the compression–tension test and (b) typical stress–strain curve from a compression test of Rohacell WF51 foam.
on comparison of pairs of digital images taken by a charged coupled device camera during the deformation progress. In order to enhance the image analysis, a random speckle pattern was applied using a spray paint on the foam core surface, underneath the indentation area, prior to testing. The initial image captured before deformation was divided into domain of sub-images. Using a digital image correlation algorithm, the corresponding position of each sub-image within the domain in the deformed image could be found providing the in-plane relative displacement field. The in-plane strain components 1ii were derived by differentiating the displacement vector field. The resolution of displacement measurements using the DSP analysis was about 1/15,000 of the total image size (approx. 60 £ 40 mm2). This gave the accuracy in measured elastic strain magnitudes of about ^ 0.1– 0.3% (absolute values of strain in %). For a crushed zone with higher strains, the strain field was quite noisy and therefore filtered, which reduced the accuracy of measured strain magnitudes to ^ 0.5– 1.0%.
at the lower boundary of the core are restricted (simulating a continuous support by a rigid foundation), and no overall bending of a sandwich beam is considered. The core is assumed to remain perfectly bonded to the face sheet during indentation and unloading. Deflection of the face sheet with bending stiffness Df is assumed linear elastic. Also, the face sheet is considered to remain undamaged. A compressive response of the foam core is elastic-perfectly plastic and the progressive core crushing mechanism is assumed. The loading – unloading sequence is modelled through the following three stages. A prescribed load or displacement is applied in the first stage simulating the indentation load. The second stage is a hypothetical state, where the load is removed and the face and core are detached. Assuming elastic bending, the face sheet deflects back and retains its original position, thus leaving an open gap between the face sheet and the foam core. In the third stage, the face and the core are ‘bonded’ together in order to find the equilibrium state and resolve the residual dent and strains in the foam core. The stages 2 and 3 are treated as purely elastic.
3. Analytical model: approach and assumptions
3.1. Stage 1—indentation loading
The analytical model considers bending of a transversely isotropic face sheet bonded to a foam core and subject to a concentrated line-load as shown in Fig. 5. All displacements
Here, a simple indentation model is employed, similar to the model described in Abrate [1] or Shuaeib and Soden [7]. The model assumes an elastic Winkler foundation for
Fig. 4. Test set-up for sandwich beam indentation.
Fig. 5. A schematic of the indentation model.
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the elastic core, and a perfectly plastic foundation for the part of the core that undergoes crushing, as schematically shown in Fig. 5. 3.1.1. Elastic solution When the load P is small, the entire foundation is elastic. The governing equation for a beam on an elastic foundation reads Df
d4 we1 þ kwe1 ¼ 0 dx4e
ð1Þ
where we1 is the face sheet deflection (the index 1 refers to stage 1) and k is the foundation modulus. In the case considered in this paper, k may roughly be estimated as k¼
ð2Þ
where Ec is the core elastic modulus and tc is the core thickness. The solution of Eq. (1) is we1 ðxe Þ ¼ e2lxe ðA sin lxe þ B cos lxe Þ þ elxe ðC sin lxe þ D cos lxe Þ
ð3Þ
Pp ¼
2Ec 1p l
ð6Þ
3.1.2. Perfectly plastic solution At the increased load, a part of the core with length 2a undergoes plastic deformation (crushing). The crushing core exerts a uniform constant reaction sp on the face sheet on the length 2a (a perfectly plastic foundation). The governing equation for the face sheet bending on a perfectly plastic foundation reads d4 wp1 þ sp ¼ 0 dx4p
ð7Þ
where sp is the plateau stress or compressive yield stress of the core material. The general solution to Eq. (7) is wp1 ðxp Þ ¼ 2
sp x4p x3p x2p þ C1 þ C2 þ C3 xp þ C4 24 3 2
The boundary conditions are now
with
l4 ¼
prior to plastic deformation can be determined as
Df
Ec tc
515
wp1 ð0Þ ¼ a;
k 4Df
The constants A; B; C and D in Eq. (3) are determined from boundary conditions which require the solution remains bounded at infinity we1 ðxe ! 1Þ ¼ 0;
w0e1 ðxe
! 1Þ ¼ 0
and that the slope at the origin be zero because of symmetry w0e1 ðxe ¼ 0Þ ¼ 0: Further, the deflection of the face sheet in the origin is
w0p1 ð0Þ
wp1 ðxp Þ ¼ a 2
sp x4p Px3p x2p þ þ C2 24 12Df 2
which gives that A ¼ B ¼ a: Thus, the elastic solution can be written as
w0p1 ðaÞ ¼ w0e1 ð0Þ
It can be seen that the elastic solution is a damped harmonic function which means that the displacement function exhibits an ‘overshoot’ at some co-ordinate xe so that there will be tensile normal stresses in the core. The contact force required to produce the initial deflection a is twice the magnitude of the shear force in the origin and can be determined as ! P d3 we1 ¼ Df !P ¼ 8Df l3 a ð5Þ 2 dx3e xe ¼0 The onset of plastic deformation (core crushing) occurs at the condition we1 ð0Þ ¼ 1p tc : By combining Eq. (2), second part of Eqs. (3) and (5), the maximum contact force
¼ xp ¼0
P 2Df
ð9Þ
A general equation for the face deflection is obtained by combining the elastic and plastic solutions of Eqs. (4) and (9). In these two expressions, there is a total of five unknowns (a; A B; C2 and P). The following four continuity equations can be used to solve for four of them wp1 ðaÞ ¼ we1 ð0Þ
ð4Þ
!
so that
we1 ðxe ¼ 0Þ ¼ a
we1 ðxe Þ ¼ a e2lxe ðsin lxe þ cos lxe Þ
¼ 0; and
d3 wp1 dx3p
ð8Þ
w00p1 ðaÞ ¼ w00e1 ð0Þ 000 w000 p1 ðaÞ ¼ we1 ð0Þ
A fifth condition is obtained by requiring that at the point of transition between elastic and plastic core, at x ¼ a; the transverse normal strain be equal to the maximum strain before crushing wp1 ðaÞ ¼ we1 ð0Þ ¼ 1p tc so that B ¼ 1p t c : Using these conditions, Eqs. (4) and (9) can be solved simultaneously for indentation force P for a given length of the plastic zone a: Also, the plastic zone a can be calculated for a given load P:
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3.1.3. Thickness of elastic and plastic core Assuming the progressive crushing mechanism of the foam, the crushed core is all in a state corresponding to the densification strain 1d ; while the core underneath the crushed zone is strained to the elastic limit 1p : In the elastic zone, x . a; the core is in an elastic state with a strain 1p at xe ¼ 0: For x ¼ a; the deflection can be written as w1 ðaÞ ¼ 1p tc
ð10Þ
The following simple approach is used to calculate the thickness of the crushed core layer for a given face sheet deformation w1 as shown in Fig. 6. It is obvious that the thickness of the core in elastic and plastic states is a function of the displacement w1 ðxÞ: They can be related through strains in the different regions as tp1 ðxÞ ¼
ðw1 ðxÞ 2 tc 1p Þð1 2 1d Þ ð1d 2 1p Þ
ð11Þ
Since w1 is a function of x; both tp1 and te1 are also functions of x: 3.2. Stage 2—removal load and detachment of face sheet from core In this stage, the load is removed and the face sheet is hypothetically detached from the core, as schematically shown in Fig. 7. After detaching the face sheet, the elastic part of the core relaxes retaining a zero strain level. The crushed core on the other hand, will remain at the residual strain level 1R ; as shown in Fig. 1. Since the face sheet was assumed to bend elastically, it will bounce back to a zero displacement level after detaching from the core, thus leaving a gap between the face and the core. At this stage, the elastic core thickness can be obtained as te2 ¼
te1 1 2 1p
ð12Þ
Fig. 7. A schematic of deformations and strains in a sandwich beam at the stage 2.
be calculated knowing the residual strains as tp2 ¼
tp1 ð1 2 1R Þ 1 2 1d
ð13Þ
3.3. Stage 3—attachment of face to core In this stage, the stage 2 is used as the base configuration and all displacements and strains are referred to the stage 2. In the third stage, an equilibrium state for which the face sheet and the core share the same displacement w3 is sought. In other words, the face sheet is given a displacement w3 ðxÞ and the core is ‘pulled up’ to the same position as schematically shown in Fig. 8. The function w3 ðxÞ for a state of equilibrium was found using an approximate method. From measurements of the residual dent after indentation or impact, it was observed that the dent has a certain length L; and a certain maximum amplitude b for x ¼ 0: The measurements also implied that the shape of the dent can be expressed approximately by a third order polynomial. Assuming that the function w3 ðxÞ can be expressed as w3 ðxÞ ¼ Ax3 þ Bx2 þ Cx þ D with the boundary conditions w3 ð0Þ ¼ b w03 ð0Þ ¼ 0 w3 ðLÞ ¼ 0 w03 ðLÞ ¼ 0
At the transition point between the plastic and elastic core, at x ¼ a; the thickness of the elastic core in this stage is te2 ¼ tc : The thickness of the crushed core in this stage can
yielding that
Fig. 6. A schematic of deformations and strains in a sandwich beam at the stage 1.
Fig. 8. A schematic of deformations and strains in a sandwich beam at the stage 3.
w3 ðxÞ ¼
2 bx 3 3bx 2 2 þ b; L3 L2
x$0
ð14Þ
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The unknown residual dent magnitude b and length L can be found using the Eq. (14) and applying the principle of minimum total energy.
517
in compression. The displacement d can thus be written as
d ¼ te3 þ tp3 2 te2 2 tp2 or substituting Eq. (17), as
3.3.1. Face sheet bending strain energy At the third stage, the bending strain energy of the face sheet can be expressed as !2 1 ðL d2 w3 Pb ¼ D dx ð15Þ 2 0 f dx2 The elastic part of the core, for a , x , L; is 1 ð L ð tc 1 ðL ðtc w3 w3 Pe ¼ s1 dx ¼ E dx 2 a 0 2 a 0 c tc tc ¼
1 ðL kðw3 Þ2 dx 2 a
Since this a Winkler type foundation model, the same stress is applied to both the elastic and the plastic part of the core. We can then write s s d ¼ te2 1e3 þ tp2 1p3 ¼ te2 þ tp2 ð19Þ Ec ER The equivalent foundation modulus kR is defined by tp2 21 s t Ec ER kR ¼ ¼ e2 þ ¼ ð20Þ d Ec ER te2 ER þ tp2 Ec
ð16Þ
The stored strain energy in the elastic/plastic part of the core, x , a; remains. The elastic part of the core with thickness te2 and zero strain at the stage 2 now changes to a thickness te3 ; while the plastic core with the thickness tp2 that changes to tp3 : The following relation for compatibility can be written using the schematic in Fig. 9. te3 þ tp3 þ w3 ¼ te2 þ tp2 þ d þ w3 ¼ tc where d is defined as the displacement of the core between stages 2 and 3. The thickness of the core layers in the elastic and plastic parts of the core is now expressed as te3 ¼ te2 ð1 2 1e3 Þ and tp3 ¼ tp2 ð1 2 1p3 Þ
d ¼ te2 1e3 þ tp2 1p3
ð17Þ
If the strain 1p3 is positive then ER ¼ ERt ; otherwise ER ¼ ERc : The point of transition, i.e. for 1p3 ¼ 0; is found by numerical iteration and is denoted xg : The total stored elastic energy in the core is then found by integration of 1 ða Pe ¼ k ðdÞ2 dx ð21Þ 2 0 R The total stored energy in the system will then be !2 1 ðL d2 w 3 1 ðL P¼ Df dx þ kðw3 Þ2 dx 2 2 0 2 a dx 1 ða þ k ðdÞ2 dx ð22Þ 2 0 R Transverse equilibrium strains are found from Eq. (19)
ERt
and tp3 t 1e3 ¼ 1 2 e3 and 1p3 ¼ 1 2 te2 tp2
1p3 ¼
ð18Þ
The modulus of the elastic layer of the core remains equal to Ec while the plastic part will have a modulus equal to ERt providing the plastic core part will be in tension and with modulus ERc if the plastic part of the core will be
d tp2 te2 þ Ec ERt
; 1e3 ¼
Ec
d tp2 te2 þ Ec ERt
; ð23Þ
0 , x , xg
1p3 ¼ ERc
d d tp2 ; 1e3 ¼ tp2 ; te2 te2 þ Ec þ Ec ERc Ec ERc
ð24Þ
xg , x , a Energy methods based on assumed displacement functions, such as that used in the analysis of unloading, provide powerful tools for solving complex problems but do not in general lead to exact solutions as the conditions of equilibrium are only approximately satisfied. 3.4. Finite element modelling
Fig. 9. Geometry used to find strains in the elastic–plastic part of the core at stage 3.
The developed analytical solutions were verified by the FE analysis of the static indentation of sandwich beams. The FE package ABAQUS was used in this study. The FE analysis was carried out for the sandwich configurations as given in Table 1. Plane strain condition was assumed in the FE analysis. Four-node bilinear plane strain elements
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(CPE4) were used. The two-dimensional mesh consisted of 1200 elements, with two elements through the thickness of each face, 20 elements through the thickness of the core, and 50 elements along the length of the sandwich beam. The FE mesh was refined towards the indentation point. This mesh was selected on the basis of a convergence study with respect to the number of elements, and the bias ratio. All degrees of freedom were constrained at the lower boundary of the model, simulating a rigid fixed foundation. Due to the symmetry conditions, only half of the sandwich beam was modelled. The indenter was modelled as a rigid cylinder with diameter of 25 mm. All degrees of freedom of the indenter were constrained, except for the translation in vertical direction (normal to the upper face sheet plane). The interaction between the indenter and the upper face sheet was modelled using a contact algorithm implemented in ABAQUS. The FE model is shown in Fig. 10. The plastic response of the core material was modelled using *CRUSHABLE FOAM and *CRUSHABLE FOAM HARDENING options in ABAQUS, version 6.3. This material model was calibrated using the test data from a uniaxial compression of the foam material. The data was given in terms of uniaxial compression yield stress versus corresponding logarithmic plastic strain. The face sheets were modelled as linear elastic. Since the sandwich structure undergoes large deformations, geometrically non-linear analysis was conducted using the *NLGEOM option. The unloading after indentation was simulated applying a reversed prescribed displacement of the rigid cylinder. The contact interaction between the face and the cylinder was computed automatically by ABAQUS. The stiffness characteristics of the volume of the core that was crushed at the loading step were kept unchanged. Thus, the FE model was expected to provide a ‘more stiff’ response for the unloading curve than observed in the experiments. Subsequently, the residual dent magnitude will be somewhat overestimated.
4. Comparison of theoretical and experimental results 4.1. Elastic response to indentation load The critical load Pcr at which core crushing first occurs was measured from the indentation tests. It was taken as a maximum load observed for the linear load – deflection response. Theoretical predictions of the critical load were obtained using Eq. (6). A comparison of experimentally measured and predicted critical loads is presented in Table 2. Theoretical and FE results are very dependent on both the correct yield strength of the core sp and on the elastic properties of the face sheet. In general, the predicted values are slightly underestimated but still show good agreement with experimental results. 4.2. Elastic– plastic response to indentation load For higher loads ð. Pcr Þ; the load –deflection response becomes non-linear owing to the core crushing. Typical load –deflection curves for two tested sandwich configurations are shown in Fig. 11. The maximum indentation (deflection magnitude) in the tests was 7 mm. At higher deflections, the onset of delaminations and fibre breakage was observed in the face sheet causing a noticeable reduction in the stiffness. The load – deflection response from the tests exhibits a ‘load dip’ at the onset of core crushing. A similar behaviour was observed from the uniaxial tests of foam specimens which show that the initiation stress (stress level at which first cell layer crushes) is slightly higher than the plateau stress, Fig. 3(b). The analytical and FE models do not account for this effect exhibiting a smooth transition from linear elastic to non-elastic response of sandwich beams under indentation loading. However, the predicted load – deflection response is in very good agreement with experimental data, Fig. 11. The size of the plastic zone (crushed core) at different load levels was measured from the indentation tests using the DSP equipment. These measurements are plotted against the indentation load in Fig. 12 along with predictions from the analytical model and FE analysis. The experimentally measured size of the crushed core zone is in good agreement with analytical predictions. The results from the FE study slightly underestimate the size of the crushed core zone. Table 2 Critical load at which core crushing was first observed and comparison of the experimental data with analytical and finite element predictions
Fig. 10. Finite element model of static indentation test.
Configuration
Pcr (experiment) (kN)
Pcr (analytical) (kN)
Pcr (FE) (kN)
Rohacell WF51 core Divinycell H60 core
1.60 1.45
1.40 1.14
1.35 1.10
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Fig. 11. Static indentation response of the sandwich beams with (a) Rohacell WF51 foam core and (b) Divinycell H60 foam core.
4.3. Residual dent Experimental measurements have shown that the amplitude of the residual dent decreases with time owing to creep or relaxation effects in the crushed foam core. Therefore, the results of the residual dent measurements are very sensitive to the time passed after the indentation test. In this study, the measurements of the residual dent were performed directly in the test machine and immediately after unloading to the zero load level, thus, measuring the instantaneous residual dent magnitude. These values were used for comparison with predictions from analytical and FE models which do not account for the effect of creep. It should be mentioned that a steady-state magnitude of the residual dent is significantly lower than the instantaneous dent by as much as 25 – 40%. Fig. 13 shows the magnitude of the residual dent measured in the tests at different maximum indentation magnitude.
The correlation between experimentally measured dent magnitudes and the analytically predicted values is rather good. In the H60 case, the analytical model underestimates the dent magnitude somewhat, whereas the agreement for the WF51 configuration is much better. As it was anticipated, the FE predictions considerably overestimate the residual dent magnitude due to very simplified and rough approach used for modelling of the unloading response. The length of the residual dent was also measured in experiments after indentation at different magnitude and compared with predictions from the analytical model, Fig. 14. The agreement between measured and predicted values was good. The FE predictions (not presented in the plot) overestimate the length of the residual dent due to aforementioned reasons. In general, the predictions of residual dent parameters (magnitude and length) are very encouraging, especially considering that the analytical approach is quite simple and based on many assumptions.
Fig. 12. Indentation load vs. half-length of plastic zone predicted and measured for (a) WF51 and (b) H60 sandwich configurations.
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Fig. 13. Magnitude of residual dent plotted against the indentation magnitude for (a) WF51 and (b) H60 specimen configurations.
4.4. Residual strains in the crushed core The DSP analysis was used for experimental evaluation of residual strains in the crushed core of sandwich beams after indentation and unloading. Fig. 15 shows the residual strains distribution in the crushed core of WF51/GFRP beam immediately after unloading and after relaxation during 10 min. The plots are presented in un-deformed co-ordinates; therefore, the residual dent is not visible. It can be observed that the residual strains along the crushed core zone are quite uniform, reducing towards the exterior of the zone. Two bands of somewhat more compressed core are visible in the plots. The analytical model presented in this study was used to obtain an estimate of residual strains in the crushed core. The residual strains were estimated using the following expression 1residual ¼ 1 2
tp3 ð1 2 1d Þ tp1
ð25Þ
where tp3 and tp1 are the thickness of the crushed core at stage 3 and stage 1, respectively. This simple estimate provided quite good agreement for residual strains measured in the middle, directly under the indentation point. Thus, the maximum magnitude of residual strains in WF51/GFRP beams was measured to be about 27% immediately after unloading. The predicted values were in the order of 30%. For H60/GFRP beams, the agreement was better; the measured and predicted values of residual strains in the middle were in the order of 12 –13%. The comparison for the residual strains in the crushed core is graphically shown in Fig. 16. At this stage, the analytical model is not valid to calculate residual strains along the whole crushed core zone. The model suggests that the residual strains increase towards the exterior of the crushed core zone that is not physical. Inherent features of the presented model may explain this discrepancy. Thus, the thickness of the crushed core at stage 1, tp1 ; is a function of forth polynomial whereas
Fig. 14. Length of residual dent plotted against the indentation magnitude for (a) WF51 and (b) H60 specimen configurations.
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Fig. 15. DSP image of residual strains in the crushed foam core of WF51/GFRP beam: (a) after unloading and (b) after relaxation (10 min). The plots are given in un-deformed co-ordinates.
the thickness of crushed core at stage 3, tp3 ; is a function of third polynomial (assumed function for the residual dent). It should be possible to overcome the problem and use the same degree polynomial function, which can improve the agreement with experimental data, however, for now it is left beyond this study.
5. Discussion and conclusions An analytical/numerical approach to modelling the indentation and unloading of sandwich beams is presented in this paper and validated by experimental results. The presented analytical model is based on an elastic-perfectly plastic Winkler type foundation and utilises the inherent progressive crushing mechanism of foams. There are several simplifications used in the model and also several facts that worth noticing that have an effect on the validity of the modelling approach. The Winkler foundation assumption contains a few rather rough approximations. First of all, transverse shear stresses are not accounted for. Secondly, the foundation modulus k (Eq. (2)) is due to its definition geometry dependent, and various definitions of the foundation modulus have been used over the years by different researchers in the pursuit of models for sandwich behaviour, e.g. local loads and wrinkling. In here, a very simple definition has been used. Thirdly, lateral deformations are considered negligible thus Poisson’s ratio is taken to be zero. The model is only valid for foam cores that exhibit constant crushing stress (perfectly plastic materials). Other material behaviour would require implementation of a hardening rule and it is left beyond the scope of this study. When modelling the progressive crushing of the core, it is assumed that the core either comprises of regions with densification strain or elastic limit strain, i.e. 1d or 1p : In reality, the strain levels will not assume discrete values,
but rather have intermediate zones of varying strain magnitudes. The results obtained herein imply that the simplified approach can be sufficient to describe the crushing mechanism of foam cores. Mechanical behaviour of foam materials is much more complex than for solid materials. Thus, for analysis presented in this study, extensive knowledge about mechanical properties and behaviour of a foam material is required. Many parameters can only be extracted by quite complex experimental procedures. The outcome of the simulation depends highly on these parameters, so great effort was made to establish, for example, Young’s modulus for compressed foam, etc. In some tests, strains of approximately 70% were measured based on displacement of the crosshead and specimen geometry. Evidently, such measurements are hard to conduct without introducing some degree of error. Although these facts contribute to the overall output error, the results obtained for two different
Fig. 16. Distribution of residual strains along the crushed core zone from the DSP analysis.
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material combinations, seem to justify the use of a Winkler type foundation. To find the equilibrium shape of the deflected face sheet, energy methods were utilised. Strain/bending energies are stipulated and the total energy of the system is minimised with respect to unknown variables b and L: Energy methods provide powerful tools for solving complex problems but do not in general lead to exact solutions as the conditions of equilibrium are only approximately satisfied. Upon unloading of indented sandwich beams, an extensive creep has been observed in experiments. The effect of creep has not been implemented in the model due to lack of material data. It must be emphasised that implementing creep as a mode of deformation is vital in future work. During the experimental phase of this study, two different methods of measuring the residual dent were used. One of the methods presented in the paper allows for measuring almost instantaneous value of the dent magnitude after unloading.
Acknowledgements The authors (Dan Zenkert and Andrey Shipsha) are grateful for financial support from the ONR Research Program on ‘Composites for Marine Structures’ through the program manager Dr Y.D.S. Rajapakse. Dr Victor Rizov (Univ. of Architecture, Civil Engineering and Geodesy, Bulgaria) is acknowledged for help with finite element analysis.
References [1] Abrate S. Localized impact on sandwich structures with laminated facings. Appl Mech Rev 1997;50(2):69 –82. [2] Thomsen OT. Analysis of local bending effects in sandwich panels subjected to local concentrated loads. In: Weissman-Berman D, Olsson K-A, editors. Proceedings of the Second International Conference on Sandwich Construction. UK: EMAS Ltd 1992. p. 417–40. [3] Thomsen OT. Theoretical and experimental investigation of local bending effects in sandwich plates. Compos Struct 1995;30:85–101. [4] Frostig Y, Barush M, Vilnay O, Sheinman I. Bending of nonsymmetric sandwich beams with transversely flexible core. ASCE J Engng Mech 1991;117(9):1931–52. [5] Schwarts-Givli H, Frostig Y. Behaviour of unidirectional sandwich panels with a bi-linear soft core—higher order approach. ASME ICEME’2000 Conference in Orlando, November 5 – 10, ASME AERO/AMD AD-vol. 62/AMD vol. 245; 2000. p. 263–76. [6] Soden PD. Indentation of composite sandwich beams. J Strain Anal 1996;31(5):353– 60. [7] Shuaeib FM, Soden PD. Indentation failure of composite sandwich beams. Compos Sci Technol 1997;57:1249–59. [8] Steeves CA, Fleck NA. Failure modes in sandwich beams with composite face-sheets and PVC foam cores. ASME ICEME’2000 Conference in Orlando, November 5–10, ASME AERO/AMD ADvol. 62/AMD vol. 245; 2000. p. 21–8. [9] Mines RAW, Alias A. Numerical simulation of the progressive collapse of polymer composite sandwich beams under static loading. Composites, Part A 2002;33:11– 26. [10] Annual book of ASTM standards. Technical standard ASTM D790-92. [11] Shipsha A, Hallstro¨m S, Zenkert D. Failure mechanisms and modelling of impact damage in sandwich beams—a 2D approach. Part I. Experimental investigation. J Sandwich Struct Mater 2003; 5(1):7–32. [12] Shipsha A, Hallstro¨m S, Zenkert D. Failure mechanisms and modelling of impact damage in sandwich beams—a 2D approach. Part II. Analysis and modelling. J Sandwich Struct Mater 2003;5(1): 33 –52.