Indentation failure analysis of sandwich beams

Transcription

1 Composite Structures 50 (2000) 311±318 Indentation failure analysis of sandwich beams A. Petras, M.P.F. Sutcli e * Department of Engineering, Cambridge University, Trumpington Street, Cambridge CB2 1PZ, UK Abstract Failure of sandwich honeycomb structures under indentation loading is considered. A failure criterion for Nomex honeycombs subjected to combined compressive and shear stresses is determined using biaxial tests. By combining this with a theoretical calculation of the stress distribution in the core due to indentation loading, found from a high-order sandwich beam theory (HOSBT), the indentation failure load of the sandwich beam due to core failure can be predicted. It is assumed that both the skin and core are elastic up to failure, which is a reasonable approximation for the GFRP skins and Nomex cores considered. Short beam 3-point bending tests are used to validate the theoretical predictions, using beams made with GFRP skins and Nomex cores with densities between 29 and 128 kg=m 3. Theoretical predictions of indentation failure load are in excellent agreement with measured values. Inclusion of shear stresses in the failure criterion signi cantly improves the predictions, correctly modelling the observed stronger behaviour of cores with a longitudinal ribbon direction. Ó 2000 Published by Elsevier Science Ltd. Keywords: Sandwich beam; Indentation; High order sandwich beam theory; Arcan test; GRRP; Nomex 1. Introduction Composite sandwich panels are widely used in industries such as aerospace or sports goods, where weight is at a premium. Indentation failure is a common problem, arising from a number of causes including impact or handling or from localised loading at joints. Among the most popular choice of materials are panels made with skins of GFRP and a core of Nomex paper and phenolic resin. Indentation failure is a particular problem for Nomex cores because of their low transverse exibility. The work presented in this paper focuses on considering failure of the core in this type of material combination, where it is appropriate to assume that the beam behaves elastically up to failure. The aircraft industry, in collaboration with sandwich panel manufacturers, has for many years conducted research to establish design methods for indentation resistant and cost e ective sandwich panels [1]. Current design codes [2,3] model indentation by taking the failure load as the product of the out-of-plane compressive strength of the core and the area over which the load is applied. However, Petras and Sutcli e [4] show that such an approach is not appropriate for many practical * Corresponding author. Tel.: +44-(0) address: mpfs@eng.cam.ac.uk (M.P.F. Sutcli e). sandwich beams. The stress applied at the skin surface is signi cantly redistributed by the skin as it is transmitted into the core, so that the area over which the load is applied is not an appropriate measure of the area of the core which takes the load. This problem can be overcome empirically by choosing a contact area so that the prediction matches the measured failure load. An a priori model of failure needs to consider the distribution of stresses in the beam due to the indentation loading. The indentation response of composite sandwich beams has been modelled assuming linear elastic bending of the top skin on a rigid-perfectly plastic foundation (the core) by Soden and Shuaeib [5,6]. Olsson and McManus [7] introduced a theory for contact indentation of sandwich panels; the model is based on the assumption of axisymmetric indentation of an in nite elastic face sheet bonded to an elastic-ideally plastic core on a rigid foundation. Despite the usefulness of such models for analysing sandwich beam indentation behaviour, they are restricted to the loading case where the beam rests on a rigid foundation. However, sandwich beams in service are generally simply supported or clamped. Hence they su er both indentation and bending loads. To model these loading conditions accurately, it is important to consider through-thickness compression of the core and the in uence of the bottom skin. In this paper, we assume that the beam can be treated as elastic up to failure, so that this e ect can be included /00/$ - see front matter Ó 2000 Published by Elsevier Science Ltd. PII: S ( 0 0 )

2 312 A. Petras, M.P.F. Sutcli e / Composite Structures 50 (2000) 311±318 Nomenclature Greek symbols d length of contact between the central roller and the top skin r cc out-of-plane compressive strength of the core r zz out-of-plane compressive stress in the core shear strength of the core s cs s x u Latin symbols A f ; C t ; D f ; E c ; R 1 ; R 2 shear stress in the core angle between the plane of the core-failure specimen and the loading direction dimensionless beam geometric and material constants (see Ref. [4] for details) b c C m L long, trans m M q x W ; W 0 x beam width core thickness Fourier coe cient due to the central applied load span of the sandwich beam longitudinal, transverse honeycomb ribbon direction index giving the wavelength of the Fourier term number of terms in the Fourier series variation in applied load on the top skin with position x contact load at central loading point, per unit width (at failure) coordinate along beam length using elastic high-order sandwich beam theories (HOS- BTs), such as those proposed by Frostig and Baruch [8] and Reddy [9]. Recent work by Petras and Sutcli e [4] applies Frostig's model to GFRP/Nomex sandwich panels. This paper develops their work further to include failure prediction. Indentation loading gives rise to both shear and outof-plane compressive stresses in the core. Failure due to these components is depicted schematically in Fig. 1. To model this, the behaviour of the core under combined compressive and shear loading is required. References to biaxial loading tests for cellular materials are not common in the literature. Gibson et al. [10] have performed a series of tests on a variety of foams under biaxial, axisymmetric and hydrostatic loading conditions. Zhang and Ashby [11] have investigated the in-plane biaxial buckling behaviour of Nomex honeycombs. More recently, Stronge and Klintworth [12] have studied the yield surfaces for honeycombs under biaxial macroscopic stresses. However, the required combination of shear and compressive loading for Nomex cores has not been investigated. The low-velocity or quasi-static impact behaviour of sandwich panels has been investigated experimentally by several authors. The lack of accepted test methods for measuring impact damage resistance of composite sandwich structures lead Lagace et al. [13] to propose a new methodology based on static indentation and impact tests. Mineset al. [14,15] have tested the dropped weight impact performance of sandwich beams with woven/chopped strand glass skins and polyester foam and aluminium honeycomb cores and have simulated the upper skin post-failure energy absorption behaviour with an elastic±plastic beam bending model. Low velocity damage mechanisms and damage modes have been investigated in sandwich beams with Rohacell foam core by Wu and Sun [16] and with Nomex honeycomb core by Herup and Palazotto [17]. In this paper, we use higher order beam theory to model the stress in the honeycomb core due to indentation loading under 3-point bending. This uses a failure criterion derived for combined shear and compressive loading of Nomex honeycomb core. Predictions are compared with measurements in which short beam sandwich panels are loaded under 3-point bending. Although this paper focuses on core failure, skin failure should also be considered in the nal beam design. The mode of failure expected for a given beam geometry can be identi ed using failure mode maps, as given by Petras and Sutcli e [18]. 2. Failure criteria for Nomex honeycomb under combined shear and compressive loading Fig. 1. Schematic of indentation failure mechanisms. In this section, we use the Arcan test rig [19,20] to investigate the failure behaviour of Nomex honeycombs under shear and out-of-plane compression loading and so derive a new failure criterion for the core. The rig consists of two pairs of plane semi-circular loading

3 A. Petras, M.P.F. Sutcli e / Composite Structures 50 (2000) 311± Fig. 2. The modi ed Arcan rig. plates with antisymmetric cutouts, as illustrated in Fig. 2. The specimen is attached to these plates via loading platens. The two pairs of loading plates are loaded by a servohydraulic testing machine through standard grips. The array of bolt holes in the loading plates allows the loading plates to be attached to the grips at di erent orientations. This allows a range of values of the angle u between the plane of the specimen and the loading direction, and hence a variation of the ratio tan u of compression to shear loads applied to the specimen. Specimens were prepared by cutting rectangular pieces from each of the available sandwich panels of four di erent densities, 29, 48, 64 and 128 kg=m 3, with a 3 mm cell size. The specimens are rectangular plates typically 46 mm long and 30 mm wide, with a core thickness of 9.4 mm. Although the aim of this test is to examine the behaviour of the honeycomb core, the skins are left on the test specimen during the test. The skins do not a ect the behaviour of the core in the rig, but they do provide a useful base for attaching the core to the rig using standard acrylic adhesive. For each core density, tests were conducted for both orientations of the honeycomb ribbon (i.e., with the long axis of the specimens coincident with the ribbon longitudinal and transverse direction). Loading was monotonic up to failure, with a constant displacement rate of 0.3 mm/min. In order to cover a wide range from nearly pure shear to nearly pure compression, tests were made at three angles of u ˆ 21:5 ; 51:5 and 81:5. Fig. 3 shows load±de ection curves for the three loading angles u for a core of density 128 kg/m 3. The de ection is measured from the cross-head displacement of the machine. Fig. 3 shows that the cores behave in a relatively brittle manner. The peak load P is used to extract the combination of normal compressive load P sin u and shear load P cos u at failure. Assuming that these loads are distributed uniformly over the external surfaces of both skins, the compressive and shear stresses r zz and s x in the core Fig. 3. Load±de ection curves for specimens of core density 128 kg/ m 3, with the three loading angles u. u is the angle between the plane of the specimen and the loading direction, so that a value of u ˆ 81:5 gives predominantly compressive loading while u ˆ 21:5 gives predominantly shear. at failure are given by P sin u= bl and P cos u= bl, respectively, where b and L are the breadth and length of the specimen. Plots of the combination of shear and compressive loading at failure are derived from tests at the three loading angles u for each core density and for both honeycomb ribbon directions, as shown in Fig. 4. The applied shear and compressive stresses are normalised by the corresponding core compressive or shear

4 314 A. Petras, M.P.F. Sutcli e / Composite Structures 50 (2000) 311±318 Fig. 4. The combination of applied shear and compressive stress at failure for Nomex honeycomb cores. Shear and compressive stresses are normalised by the corresponding core strength. The dashed line corresponds to the linear failure criterion given by Eq. (1). strength r cc or s cs, taken from the manufacturer's data sheet [2] and given in Table 1. Except for the honeycombs with density 128 kg=m 3, the failure envelopes are well approximated by the linear failure criterion r zz r cc s x s cs ˆ 1; 1 Table 1 Material properties of Nomex Density kg=m 3 r cc MN=m 2 s cs MN=m 2 Longitudinal Transverse which is included as dashed lines in Fig. 4. The inconsistency observed in the honeycombs with a core density of 128 kg=m 3 is due to a di erence between our measurements and the data sheet value of failure stress under pure shear loading. This does not signi cantly a ect the accuracy of our calculations. 3. Failure analysis using high-order sandwich beam theory (HOSBT) In this section, the failure criterion established for the core in the previous section is combined with a model of the stress distribution in the core to derive a failure model for indentation loading. The HOSBT described by Frostig and Baruch [8] is used to model the core stresses. The details of how this can be applied to 3- point bending of sandwich beams is described in detail by Petras and Sutcli e [4] and only an outline is given here. In their work, Petras and Sutcli e [4] show that it is essential to model the local de ections under the indentor to make a sensible estimate of the stress distribution in the core, and that HOSBT does this reasonably well. `High-order' refers to the non-linear way in which the in-plane and vertical displacements are allowed to vary through the height of the core. The core vertical displacement is assumed to have a quadratic

5 A. Petras, M.P.F. Sutcli e / Composite Structures 50 (2000) 311± variation in the through-thickness direction. Other core displacements also vary in a non-linear way, with the exact variation being expressed in terms of a Fourier series. This contrasts with simple beam theory, where the core in-plane displacements are assumed to vary in a linear way through the depth, and the out-of-plane displacements are assumed to be constant. These highorder variations allow modelling of the more complicated changes in core geometry which occur at loading points. Thus the basic assumptions of HOSBT are: The shear stresses in the core are uniform through the height of the core. The core vertical displacement varies as a quadratic polynomial in the through-thickness direction, allowing the core to distort and its height to change. The core is considered as a 3-D elastic medium, which has out-of-plane compressive and shear rigidity, but negligible resistance to in-plane normal shear stresses. To simplify our calculations we assume that the indentation line load W is applied to the beam of length L and core thickness c as a uniform pressure over a given width d ˆ 1 mm at the midspan of the beam. This assumption will be good for sandwich panels typically used in industry, and it's validity is discussed in detail by Petras and Sutcli e [4]. The variation of applied load distribution q x with position x on the top skin is described by a Fourier series of M terms The line load W 0 x which would cause failure in the core at a position x along the beam is found by substituting the variation in shear and compressive stress in the core± skin interface (Eqs. (5) and (4)) into the core failure criterion, Eq. (1) r zz x s x x ˆ 1: 6 r cc s cs The location of rst failure and the corresponding critical line load W 0 is found from the minimum value of W 0 x. To examine the importance of including the shear Fig. 5. Experimental setup. q x ˆXM mˆ1 C m sin mpx L 2 with Fourier coe cients C m given by C m ˆ 4W mpd sin mp sin mpd ; 3 2 2L where m is an index for the wavelength of the Fourier term. For the failure analysis, we need to know the stress eld in the core-top skin interface; i.e., the out-of-plane normal stresses r zz and s x. Petras and Sutcli e ([4], equation (13)) show that the normal stress component can be expressed in terms of the equivalent Fourier series as r zz x ˆXM E c mˆ1 C m cmp 2L D f E c A f C t R 1 R 2 sin mpx L ; 4 where the bold symbols in the above equation are combinations of the material and geometric properties of the beams, as detailed by Petras and Sutcli e [4]. The equivalent Fourier series for the shear stress component is given by Petras [21] as s x x ˆXM mˆ1 C m A f C t cos mpx R 1 R 2 L : 5 Fig. 6. Variation of line load and midspan core compression with top skin de ection for a sandwich beam with a core density of 29 kg=m 3, loaded by a central roller of diameter 6 mm. Legend: ±±± longitudinal and (± ± ±) transverse honeycomb ribbon direction.

6 316 A. Petras, M.P.F. Sutcli e / Composite Structures 50 (2000) 311±318 stresses, a failure line load is also predicted as when the maximum compressive stress in the core (directly under the indentor) equals the core compressive strength. 4. Experimental measurements of indentation failure To validate the theoretical model of indentation failure, 3-point bending tests were performed on short sandwich beams of width 30 mm and length 70 mm. Panels had the same 9.4 mm thick cores tested in Section 2, with nominal densities 29, 48, 64 and 128 kg=m 3,and 0.38 mm thick skins of cross-ply GFRP laminate. Further details of the material properties are given in [4]. The experimental setup is depicted in Fig. 5. The specimens were loaded under 3-point bending using two outer rollers spaced 60 mm apart and a central roller of three di erent diameters (6, 10 and 20 mm). The central roller was displaced relative to the outer rollers at a rate of 0.5 mm/min. A clip gauge on one side of the specimen recorded the midspan core compression during loading while a video camera recorded images of the other side of the beam. Fig. 6 plots typical variations with the top skin de- ection of the applied line load and midspan core compression (i.e., the di erence in out-of-plane displacements on the top and bottom of the beam measured at the midspan). These results are for beams with a core density of 29 kg=m 3, loaded by a central roller of diameter 6 mm. Separate curves for longitudinal and transverse ribbon directions are included. Very similar curves are observed for the whole range of core densities, ribbon orientations and roller diameters. Fig. 6 shows that there is signi cant core compression, explaining the need for a high-order beam theory to capture the local indentation response. The variation of applied load and core compression with top skin de- ection is almost linear up to the peak load, which is taken as the failure load. This justi es the use of an elastic model of the beam to predict failure. The core compression is greater for beams with a longitudinal than with a transverse honeycomb ribbon direction, at a given line load, because of the higher shear sti ness of the Nomex honeycomb with this ribbon orientation. Fig. 7(a) and (b) illustrate the portion of the sandwich beams under a central roller of diameter 10 mm, just before failure and well after the peak load is reached, corresponding to a top skin de ection of approximately 2 mm. The cores have densities of 29 and 64 kg=m 3, for Fig. 7(a) and (b), respectively. For the low density core, Fig. 7(a), there is signi cant shear in the core and widespread elastic buckling of the cell walls before failure. Fig. 7(b) shows that the high out-of-plane sti ness of the high density core tends to prevent deformation of the core away from the loading point. The core Just prior to peak load Well after the peak load (a) Just prior to peak load Well after the peak load (b) Fig. 7. Photographs of a sandwich beam under a 10 mm diameter central roller just prior to peak load and well after the peak load. The core density is (a) 29 kg=m 3 and (b) 64 kg=m 3 with a longitudinal ribbon direction.

7 A. Petras, M.P.F. Sutcli e / Composite Structures 50 (2000) 311± Conclusions Fig. 8. Comparison between predictions (lines) and experimental measurements (symbols) of the indentation failure line load W 0. eventually fails by plastic buckling of the cell walls directly under the indentor. The complex deformation elds in the core explain the need for high order beam theory and inclusion of both shear and compressive stress components when predicting indentation failure. 5. Comparison between predictions and measurements of indentation failure load Fig. 8 shows a comparison between the predicted and measured indentation failure loads for four di erent core densities, two core ribbon directions and three roller diameters. Theoretical predictions are included, which either consider both shear and compressive stresses in the core, or just the compressive component. Predictions are in good agreement with measurements. Measured failure loads do not vary signi cantly with the indentor size, as indicated by the results of Petras and Sutcli e [4], who show that the core stresses for this material and beam con guration are relatively insensitive to the details of the load distribution. This also justi es the assumption made in the theoretical modelling that the load is applied as a uniform pressure over a xed width d. Fig. 8 shows that predictions using the failure criterion which includes shear and compressive stresses in the core are signi cantly better than using a criterion which considers only the compressive stresses. Moreover, the combined failure criterion successfully predicts the higher indentation resistance of sandwich beams with a longitudinal honeycomb ribbon direction. A systematic approach has been developed to predict failure of sandwich honeycomb structures under indentation loading. It is assumed that both the skin and core are elastic up to failure, which is a reasonable approximation for the GFRP skins and Nomex cores considered. A failure criterion for Nomex honeycombs loaded simultaneously by out-of-plane compressive and shear stresses has been determined using biaxial tests with an Arcan rig. A linear dependence on the compressive and shear components is found to approximate well the measured response. HOSBT is used to predict the compressive and shear stress distribution in the sandwich beam core under indentation loading. By combining this stress distribution with the failure criterion determined from the biaxial tests, the failure load of the sandwich beam due to core failure can be predicted. This criterion can be used to model failure as the failure mode switches from pure core crushing to pure core shear, depending on the core and skin material properties and the beam geometry. Skin failure should also be considered separately, following the approach suggested by Petras and Sutcli e [18]. Short beam 3-point bending tests are used to validate the theoretical predictions. Measurements of the core compression under the central indentor demonstrate the need for HOSBT in predicting the core deformation, while video images illustrate the importance of core shear, particularly for low density cores. Theoretical predictions of indentation failure load are in excellent agreement with measured values. Inclusion of a shear component in the failure criterion signi cantly improves the predictions. Moreover, this model is able to predict the observed stronger behaviour of cores with a longitudinal ribbon direction. Acknowledgements The authors are most grateful to Hexcel Composites for their help in supplying materials and for the technical assistance of Mr. Nigel Hookham and Mr. Peter Clayton. Thanks are due to Messrs. Alan Heaver and Simon Marshall for their valuable contribution to the experiments and to Prof. Norman Fleck for his advice. The authors acknowledge with gratitude the nancial support of the Greek Government and the US O ce of Naval Research (grant number J-1916). References [1] Armstrong KB. Cost-e ective design of indentation resistant sandwich panels for aircraft oors and other purposes. Tech. Rep. EEA.S , BOAC, December [2] Ciba Composites, Duxford, England, Honeycomb Sandwich Design Technology, August 1995.

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