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Progressive damage analysis of tee joints with viscoelastic inserts
Composites: Part A 32 (2001) 641±653
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Progressive damage analysis of tee joints with viscoelastic inserts J.I.R. Blake a, R.A. Shenoi a,*, J. House b, T. Turton b a
Fluid Structure Interactions Research Group, School of Engineering Science, University of Southampton, High®eld, Southampton, SO17 IBJ, UK b Mechanical Sciences Sector, DERA, Farnborough, Surrey, GU14 OLX, UK Received 14 April 2000; revised 8 September 2000; accepted 11 October 2000
Abstract The purpose of this paper is to investigate the static structural response of a new type of composite tee joint containing a viscoelastic insert. The introduction of this material has proven bene®ts in terms of noise and vibration attenuation across the joint. The effects of introducing this new material on the structural response of the joint are numerically examined by using a progressive damage model. Application of this method allows the initiation and progression of failure and ultimate failure load to be predicted. Experimental results show good qualitative and quantitative agreement with the predictive damage model. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: B. Stress transfer; B. Delamination; C. Finite element analysis; D. Mechanical testing; Failure criteria
1. Introduction Large ®bre-reinforced plastic (FRP) structures, by necessity of design and production constraints, have a number of in-plane and out-of-plane joints in their topologies. In-plane joints have been the subject of attention of many researchers, for example, Godwin and Matthews [1] and Matthews et al. [2], with attention being focused on analytical treatments, numerical analyses, experimental studies, failure criteria and material aspects. Arguably, the more dif®cult problem is the one that pertains to load transfer between two orthogonally placed members meeting at a joint. The joint, an example of which is illustrated in Fig. 1, is formed by placing laminated strips of reinforcement cloth on both sides of the joint (boundary angle or overlaminate). The reinforcements are usually alternate layers of woven roving (WR) and chopped strand mat (CSM). The numbers of plies or laminae will depend on the required stiffness for the joint. The resulting gap formed between the cloth and plates is ®lled with an appropriate ®llet resin, i.e. one that is compatible with the cloth material and is generally one with a high yield strength. The weakness in this case is due to the lack of reinforcement across the connected surfaces and through the occurrence of stress concentrations associated with joint geometry and production considerations. * Corresponding author. Tel.: 144-1703-592316; fax: 144-1703593299. E-mail address: [email protected] (R.A. Shenoi).
There is now a growing body of literature on the behaviour of laminated tee joints. Early work concerned theoretical modelling using simple approaches with plane strain [3] and plane stress [4]; these corresponded to marine and aircraft applications, respectively. These efforts were rather restrictive owing to the relatively immature ®nite element analysis (FEA) capabilities at that time. More recent effort, incorporating layered (®nite) elements and a variety of failure criteria, has successfully characterised through numerical analyses both single skin [5] and sandwich [6] tee-joints under representative static loadings. This work has been supplemented by experimental programmes studying failure mechanisms [8] and stress patterns [9]. The work has been extended to studying the longterm effects of such joints under repetitive, cyclic loadings [10] including inception and progression of failure as well as approaches to life modelling. All such work has formed the basis of writing a procedure for the synthesis of design variables for typical single skin tee joints in ship and civil construction [11]. What is now being envisaged is to better utilise the joint in a dynamic response mode. One way to achieve this is to investigate the in¯uence of high strain-to-failure, viscoelastic materials. It has been shown that the use of such materials could lead to better energy absorption capabilities in structures. Preliminary work [12] has demonstrated that a tapered viscoelastic layer placed between a GRP beam and a steel supporting substrate can produce a signi®cant absorption of vibrational energy. The purpose of this paper is to consider the effects of
1359-835X/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 1359-835 X(00)00 158-5
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Fig. 1. Basic constituents of viscoelastic inserted tee joint and dimensions (mm).
placing such a material in a tee joint on its strength and stiffness characteristics.
that the use of a viscoelastic wedge absorbs compression wave energy, i.e. little energy was re¯ected at the joint interface and little was transmitted across. Furthermore, the radiated noise characteristics were much lower than those of conventional butt technology. Whereas the dynamic behaviour of the material and structure have been and are being assessed elsewhere, there is currently no knowledge of the strength and load-bearing capability of structures incorporating such materials. Furthermore, although coupon tests can and do indicate innate strength of the material itself, there is a weakness related to the manner in which it behaves within a structure. It is important therefore to understand load transfer, damage initiation and progressive build-up of damage in generic structures, such as tee joints, incorporating these viscoelastic materials.
3. Progressive damage model 2. Concepts in energy absorbent joints
3.1. Background
There are many instances in structural applications where noise and vibration transmission is unwelcome. One such example can be seen in sonar domes. Once installed on an operational boat, hydrodynamic ¯ow and supporting structural induced vibrations cause the dome to vibrate thus radiating noise and interfering with sonar sensor response. The introduction of a composite viscoelastic joint produces a vibration sink that can absorb ¯ow-generated and structure-borne noise within the dome [12]. In conventional FRP, the resins tend to have a relatively high modulus with as high a glass transition temperature Tg as possible. When matrix resins of this type are used with ®bre reinforcements, the result is a composite with high speci®c stiffness but relatively low intrinsic damping. Indeed all efforts to seek matrix resins with higher Tg (particularly in the aircraft industry) only exacerbate the problem, resulting in composites with lower and lower damping. This paper is aimed at considering the use of resin matrices with high intrinsic damping and damage tolerance, which can be optimised for speci®c frequencies and temperatures. Such composites may ®nd applications in the fabrication of new and novel structural connection concepts. House [12] describes previous work on acoustic interaction at a simple joint. It is shown that high vibration re¯ection losses can be obtained at a GRP±steel interface provided the joint input impedance is matched. This is generally coupled with a low transmission loss but at the expense of a good re¯ection loss. In all these cases, vibration energy is not lost or absorbed but merely redirected or altered by mode conversion. An alternative approach is to provide a joint that actually absorbs the energy. Current research shows that the introduction of a vibrational sink such as a viscoelastic polymeric interlayer between the GRP and steel signi®cantly attenuates the transmissibility of noise and vibration. House [12] concluded
Structural response of composite structures with intrinsic ¯exibility and non-linear material stiffness characteristics requires a non-linear ®nite element (FE) analysis. A linear analysis would assume that, despite the material failure at a location inside the joint, the global properties of the joint are not compromised in any way and the joint can be subjected to further increasing loads. In reality, there are many mechanisms for failure within the composite joint, that whilst not necessarily leading to ultimate failure at a given load, will affect the joint characteristics as the load is increased. There has been much experimental and numerical work on the failure behaviour of composite laminates. However, these investigations have concentrated on in-plane load conditions of tension, compression and shear [13±17]. Out-of-plane loading of laminated plates has received much less attention due to the material and geometric non-linearites but some important investigations can be found in Reddy and Reddy [18], Kam and Sher [19], Echaabi et al. [20], Tolson and Zabaras [21] and Padhi et al. [22]. The progressive damage of out-of-plane composite structures other than laminated plates has not been investigated in an entirely rigorous manner. Phillips [23] investigated the progressive damage of tee joints and top-hat stiffeners by considering the manual insertion of delaminations within the structure. At each load step, a visual inspection of the stress distributions determined the applicability and subsequent location of an inserted delamination. This work produced an insight into the failure mechanism of the two types of structure but was not accurate in simulating the structural response. Padhi et al. [22] successfully used FE modelling incorporating a progressive damage subroutine. This automated progressive damage modelling ®rst determines the load and
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Table 1 Material properties for the polyester/woven roving glass laminate in the ¯ange, web and overlaminate in the tee joint
single tensor polynomial failure criterion such as that proposed by Tsai. Failure is assumed to occur if the following condition is satis®ed:
Property
Value
Fi s i 1 Fij s i s j 1 Fijk s i s j s k 1 ¼ $ 1
Ex Ey Gxy Density Poisson ratio
17 100 MPa 7400 MPa 4400 MPa 1650 kg m 23 0.15
The two-dimensional form of the above polynomial is expressed as:
location at which a material in the structure ®rst fails. From that stage onwards, the failure at that location is included in subsequent analyses. The progression of damage is therefore determinable and so too the ultimate collapse load. Progressive failure analysis is based on the assumption that the damaged material can be substituted with an equivalent material with degraded properties. This can be accomplished in two principal ways: 1. Total discount method: The stiffness and strength of a failed material ply is reduced to zero. This approach may lead to an underestimation of the laminate strength because it does not recognise that ply failure is localised and that the remaining stiffness of the laminate ply is not necessarily zero. 2. Limited discount method: The reduction of stiffness depends upon the failure mode in action. For ®bre failure, the longitudinal stiffness is degraded, whereas for matrix failure, zero stiffness and strength are assigned to the failed ply for the transverse mode. The actual method used is the limited discount or stiffness reduction approach and is based on the work of Chang and Chang [13]. The method for stiffness reduction is simple but effective. For matrix cracking at a material integration point, the transverse modulus Ey and Poisson's ratio n yx are reduced to zero. However, the longitudinal modulus Ex and the shear modulus Gxy remain unchanged. When ®bre±matrix shearing is predicted at a material point, the transverse modulus Gxy and Poisson's ratio n yx are reduced to zero. However, the longitudinal modulus Ex and transverse modulus Ey remain unchanged. If ®bre failure is detected, then the material is deemed to have lost complete stiffness at the integration point.
1
F1 s 1 1 F2 s 2 1 2F12 s 1 s 2 1 F11 s 12 1 F22 s 22 1 F66 s 62 $ 1 2 In the expressions, the notations, s 1, s 2, s 6 s 6 s 12 are the in-plane stresses in the material coordinate directions. The Fij terms are the failure indices and are weighted according to the importance of individual stress components. There are different methods for deriving the failure indices; in this paper only the Tsai±Hill failure criterion is used. The failure indices for the Tsai±Hill criterion are F1 0; F11
1 2; X
F2 0; F22
F12 2
1 2; Y
1 ; 2X 2
F66
1 2 SC
3
where if s 1 . 0 then X XT, otherwise X XC and if s 2 . 0 then Y YT, otherwise Y YC.XT, XC, YT, YC and SC are strength parameters (ultimate tensile strength and ultimate compressive strength in the X, Y and XY directions). 3.3. Progressive damage assessment For the interactive polynomial criterion, if failure occurs the following expressions are used to determine the failure mode: H1 F1 s 1 1 F11 s 12 H6 F66 s 62
H2 F2 s 2 1 F22 s 22
4
The largest Hi term is selected as the dominant failure mode and the corresponding modulus is reduced to zero. Thus, H1 corresponds to ®bre failure, H2 corresponds to matrix cracking and H6 corresponds to ®bre±matrix shearing failure. The stiffness reduction method is applied and depending upon the mode of failure, the material properties are degraded accordingly.
3.2. Polynomial failure criteria
4. Numerical modelling
The next question is how to predict failure? In general, failure criteria can be categorised in two classes: independent and interactive (or polynomial). The former is simple to apply and gives the mode of failure, but it neglects the effect of stress interactions in the failure mechanism. The latter includes stress interactions in the failure mechanism, but it does not give the mode of failure. Most failure criteria for composite materials can be expressed in terms of a
4.1. Modelling basis The numerical model has been constructed and analysed using abaqus 5.8. The model is two-dimensional and made up of 3691 four-noded CPS4 elements. A plane stress analysis is used with specimen thickness of 150 mm. The joint geometry is given in Fig. 1 and the material properties in Table 1.
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Fig. 2. Adopted mesh for ¯exible tee joint (detail of overlaminate radius and web to overlaminate bond shown in inset). Viscoelastic insert mesh not shown. Loading conditions also shown.
A large number of ®nite element models have been constructed and tested in order to optimise the ®nal progressive damage model. The progressive damage model in essence requires the stiffness degradation of localised points within the structure to represent damage. The less re®ned the ®nite element mesh, the more signi®cant the effect of localised stiffness reduction on the global strength of the joint. Therefore while a coarse mesh reduces computational expense, its adverse effect on the accuracy of the progressive damage model may be signi®cant. The ®nal adopted model thus re¯ected these two opposing considerations. Previous experimental and numerical studies (e.g. [5,23] showed that large through-thickness stresses are expected in
Fig. 3. Stress±strain curves comparing relative stiffness of the standard crestomer inset material used in in-service tee joints and the viscoelastic insert material.
the overlaminate prior to delamination. The density of elements is increased in the overlaminate, focussing the attention of the structural response and aiming to improve the accuracy of the progressive damage model. Each ply of the overlaminate is represented by one element of thickness 0.61 mm. Fig. 2 shows the adopted mesh layout. 4.2. Loads, material properties and boundary conditions Fig. 2 describes the boundary conditions applied in the numerical models. The numerical model is constrained at nodes on the upper surface of the ¯ange, 500 mm apart. Both nodes are prevented from translation in the y-direction (plane of the web) but to restrict a rigid body motion, one of the nodes is restricted in the x-direction also (in the plane of the ¯ange). The progressive damage model increases the displacement from 0 to 10 mm in steps of 0.1 mm. The displacement rate is considered small and so the response of the joint is quasi-static. Constantly increasing displacement as opposed to increasing load is used in order to attempt to pick up any stress relief. Linear material properties have been used for the overlaminate, the web and the ¯ange. The viscoelastic insert exhibits nonlinear stress±strain characteristics and is modelled herein using a bilinear stress±strain curve illustrated in Fig. 3. This particular viscoelastic material was produced and tested at DERA Farnborough and is custom made for noise attenuation at speci®c frequencies and temperatures. The abscissa is foreshortened for clarity of the initial stiffness of this particular elastomer, but in®nite strain is considered for numerical analysis.
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4.3. Incorporation of progressive damage model
5. Validation and results
The procedure for progressive failure analysis adapted in this context is given below. This is programmed into a fortran standard user subroutine USDFLD, incorporated and used in the abaqus 5.8 processor. This routine allows the user to de®ne material properties as functions of the ®eld variables at a material point. The material properties of the joint were de®ned to be dependent upon three ®eld variables. The ®rst ®eld variable was the contribution of s 1 (H1) towards the failure index. The second and third ®eld variables were the contributions of s 2 and s 12 (H2 and H6, respectively) to the failure index:
5.1. Experimental results
1. At the load step, a geometric non-linear analysis is performed until a converged solution is obtained. 2. The stresses at every integration point are evaluated. 3. The failure polynomial is determined and if larger than 1.0.¼ 4. ¼the contribution of each stress component towards the failure index (Hi) is computed and the stress component, which contributes the maximum is identi®ed. 5. Depending on the largest Hi term, the failure mode is determined and the corresponding material properties are degraded accordingly. 6. A geometrically non-linear analysis is again performed for the degraded structure to re-establish equilibrium. If no more failure is detected then¼ 7. ¼the next load step is applied, 8. ¼otherwise the structure is degraded again according to steps 3±6. At any load step, a ply failure load is the load at which the failure index (Fis i 1 Fijs is j) reaches a value of unity at any material integration point. At some point in the analysis, the global stiffness becomes zero or negative and this is taken as the inability of the structure to support additional load. This location is identi®ed as the ultimate failure load. Global failure can also be interpreted based upon the degree of failure, mode and location that exists. This interpretation is important if there is likely to be considerable residual strength of a restrained substructure that remains loaded. For example, joint debonding does not lead to full structural failure as there still exists residual stiffness from the ¯ange, however the joint itself is now considered useless. The numerical models are based upon continuum mechanics and as such do not consider the discontinuity produced by physical debonding and the related redistribution of stress. Therefore, whilst the predicted model response shows the existence of positive global stiffness, in real terms the predicted amount of material failure in speci®c regions would have produced complete redundancy of the physical structure. This method of interpreting speci®c failure to cause structural redundancy was employed in the experimental tests.
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The numerical model is validated using the results of the experiments upon the physical joint. Due to the scarcity of these unique joints and the cost of producing the viscoelastic insert material and manufacturing the joints, only one destructive experiment was performed. The physical specimens were fabricated in Vosper Thornycroft shipyard. The web and ¯ange were cut from a single laminated plate produced using a resin infusion process. The viscoelastic insert was bonded on top of the ¯ange underneath the web. The cured surfaces of the web and ¯ange were abraded before the overlaminate was laid up either side of the joint. The overlaminate material was then resin infused. The specimens were loaded in a standard three-point bend described in Fig. 2. This type of loading is considered to be representative of an UNDEX 1 shock event experienced in marine environments [24]. The hull plating either side of the stiff joint is subjected to large negative pressures caused by shock bubble implosion. Rollers with a 30-mm diameter were used at the contact points on the specimen, 500 mm apart. A displacement was applied at the structure's centreline and the reaction load at the point of application was measured against increasing displacement. The tests were recorded on video so that the failure events from the video could be related to the load against displacement curve. Fig. 4 describes the experimental results of load versus displacement and shows a series of load drop-offs when substantive failure leads to progressive increase in structural ¯exibility. The events corresponding to the various load drops are listed in Table 2. Fig. 5 illustrates the ®nal failure of the joint and position of initial failure. The initial response of the joint is fairly linear upto x (Fig. 4 although there appears to be some stiffening between 2 and 3 mm de¯ection. Correlation of the response curve with the video record shows that at a reaction load of 7 kN, location x, a crack has initiated in the bond between the overlaminate and the web at the base of the web, shown by the annotation in Fig. 5. This cracking accounts for the load drop at x. Between x and y, the joint response is again linear in nature. From the video record, the crack propagates up the overlaminate to web bond until the overlaminate detaches completely from the web at y. This marks the point of structural redundancy, and therefore ultimate failure, even though the joint can take additional load through the deformation of the ¯ange. 5.2. Validation of global structural response Comparisons between numerical and experimental results are made with regard to load versus displacement curve and the initiation and progression of failure and 1
Underwater explosion.
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Fig. 4. Comparison of experimental and predicted load±displacement results. The letters correspond to failure events shown in Table 2.
ultimate failure load. Numerical and experimental results of load against displacement are shown in Fig. 4. Fig. 4 and Table 2, which correlates the failure events in Fig. 4 with physical observations, demonstrate that the predicted response of the tee joint agrees favourably with the experimental results. Initial stiffness of the numerical model is comparable to the experimental result. The predicted stiffness up to the large stress relief at event a is fairly linear. However, unlike the experimental results, which exhibit a small amount of stiffening between 2 and 3 mm de¯ection, there appears to be a slight increase in joint ¯exibility and originating at around 0.5 mm. The large load drop at a is over-predicted in terms of joint de¯ection, but the reaction load at which this drop occurs agrees well. The trend of the numerically computed results of load versus de¯ection between a and b compares favourably with the experimental results. However the ®nal failure load and de¯ection is, at ®rst sight, in poor agreement with experiment.
5.3. Internal load transfer and failure patterns To understand the global structural response predicted by the numerical models, it is necessary to investigate the causal factors namely internal load transfer mechanisms and failure patterns. The slight departure of the initial response of the joint from linear behaviour, originating at 0.5-mm de¯ection, can be explained in Fig. 6, which describes the amount of matrix cracking within the structure at this de¯ection. It is evident that at this very low de¯ection of 0.5 mm, there is already a small amount of matrix cracking occurring within the overlaminate to web bond, illustrated by the proportion of red colour in this region. The asymmetry in the distribution of damage is thought to arise from numerical rounding within the mesh generation. Any asymmetry of the generated nodal coordinates in the yz-plane (Fig. 2 Ð z-direction is out of the paper) would lead to asymmetric loading and therefore predicted damage.
Fig. 5. Final failure of viscoelastic inserted joint, event y in Fig. 4.
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Fig. 6. Distribution of damage (red) in terms of matrix cracking at a de¯ection of 0.5 mm.
Fig. 7 demonstrates the through-thickness stress distribution and Fig. 8 the shear stress distribution within the FRP of the joint at 0.5 mm de¯ection. Increasing the de¯ection from 0.4 to 0.5 mm results in an increase in stress magnitude greater than that observed in prior increments. The distribution of through-thickness stress and shear stress remains fairly constant, with higher stress gradients at the outer plies of the overlaminate radius. The deviation from the linear behaviour of the numerical model's response in the ®rst few increments corresponds to the inception of matrix cracking in the overlaminate to web bond demonstrated in Fig. 6. The load at which the large stress relief occurs in both the numerical model (point a) and in the experimental model (point x) agree well. However, the numerical model over predicts the de¯ection at which this event occurs. The reason for the stress relief can be seen in Figs. 9 and 10, which shows the distribution of matrix cracking prior to (Fig. 9 and after (Fig. 10 the event. The distribution of damage is represented by the red
colour in the above ®gures. Prior to event a, there is already a considerable amount of matrix cracking within the overlaminate to web bond, but the largest rate of damage occurrence happens at the de¯ection corresponding to a reaction load of 6.78 kN. This sudden increase in damage may be analogous to the inception of a crack in the base of the overlaminate to web bond seen experimentally. In the region x to y in Fig. 4, the crack between the web and the overlaminate in the physical specimen propagated up the web away from the ¯ange leading to complete debonding at y. The numerical models predict a different failure pattern between the equivalent points a and b. Figs. 11±13 show the distribution of through-thickness stress and shear stress in the FRP and the proportion of matrix cracking predicted at a de¯ection of 4.9 mm. Although there is little change in the distribution of stresses and negligible increase in shear stress magnitude in the interval of 4.8±4.9 mm, the through-thickness stress increases markedly from a de¯ection of 4.8±4.9 mm over stress increases in previous increments. This sudden increase coincides with the initiation
Fig. 7. Distribution of through-thickness stress in the FRP at 0.5 mm.
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Fig. 8. Distribution of shear stress in the FRP at 0.5 mm.
and propagation of matrix cracking through the outer plies of the overlaminate radius illustrated in Fig. 13. The through-thickness stress concentrations, demonstrated in Fig. 11, are on the right-hand side of the joint and yet Fig. 13 describes the initiation and propagation of matrix failure on the left-hand side. The largest stress contributor to the failure polynomial determines the dominant failure mode. However, due to the interactive nature of the stress contributions upon the failure polynomial, the in-plane stress contribution (which is much greater on the left-hand side of the joint) affects the location of predicted matrix failure. Absolute shear stress values are comparable either side of the joint.
The de¯ections of 4.8±4.9 mm correspond to the minima in the predicted load±de¯ection graph (Fig. 4. After this minima, the incidence of matrix cracking increases throughout the overlaminate radius accompanied by ®bre±matrix shear failure whilst the incidence of failure within the web to overlaminate bond region is reduced. Consequently, the ®nal failure load and mode predicted by the numerical model is in disagreement with the experimental results with Figs. 14 and 15 describing the predicted ®nal distribution of damage. However, observation of the failure pattern within the predicted model leads to conclusions for the point of structural redundancy. For example, Figs. 16 and 17 show that there is total matrix and shear failure throughout the
Fig. 9. Distribution of damage (red) in terms of matrix cracking prior to event a in Fig. 4.
Fig. 10. Distribution of damage (red) in terms of matrix cracking after event a in Fig. 4.
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Fig. 11. Distribution of through-thickness stress in the FRP at 4.9 mm.
bond region, in real terms most likely describing detachment of the overlaminate from the web. This occurs at a de¯ection of 6 mm, which corresponds to a reaction load of 7 kN, which is much lower than the predicted ®nal failure load and in closer agreement with experimental observations. 5.4. Discussion The progressive damage model has provided good qualitative and quantitative agreement with experimental results. However, it is important to treat the results with caution given the limited number of experiments that could be performed. The conclusions drawn are based on the supposition that the experimental data is representative of this speci®c joint design incorporating these materials and under the described form of loading. The con®dence in
the numerical modelling is re¯ected by the agreement with the available experimental data for these speci®c joints (bearing in mind the above supposition) and between the progressive damage methodology and a set of experimental tests, carried out simultaneously with the tee joints, on top hat stiffener sections [25]. To improve the modelling further, the material strength limits require re®ned de®nition. Phillips [23] demonstrated the importance of tolerance speci®cations for published material properties on numerical modelling. Furthermore, production processes rarely result in two specimens ever being the same, thereby providing some uncertainty as to which material strength limits to employ. This uncertainty existed in the numerical modelling of the specimen described herein. As described in Section 4.1, the adopted ®nite element mesh is a compromise between computational expense and
Fig. 12. Distribution of shear stress in the FRP at 4.9 mm.
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Fig. 13. Distribution of matrix cracking (red) at 4.9 mm de¯ection.
accuracy. The number of elements was increased in the progressive damage model until a state of convergence in terms of load±displacement result was reached. The only effect of increasing the number of elements past this point was to increase the amount of CPU time for solution. Therefore, to increase correlation between experiment and theory in the initial location of failure, the numerical strength limits for interlaminar tensile strength (ILTS) and shear strength (ILSS) were reduced in the bonded regions. Table 3 describes the assumed strengths used in the progressive damage model throughout the FRP material and the reduction of these limits in the overlaminate to web and overlaminate to ¯ange bond regions. The necessity for strength reduction in these regions highlights a possible structural de®ciency caused in the production of the specimens through lack of resin infusion. Observation of the failed physical specimen re¯ected this
Table 3 Comparison between assumed material properties through the joint (`Bond' failure limits relate to web-to and ¯ange-to overlaminate failure limits. `FRP' relates to strength limits applied throughout the rest of the FRP material) Failure limits (MPa) In-plane
ILTS
Shear
Insert
FRP
Bond
FRP
Bond
FRP
Bond
261
261
31.5
7.5
50
20
4.65
hypothesis; there was little ®bre breakage in the bonded area and a clean glassy surface remained on each of the join faces. Consequently, the progressive damage model can reinforce con®dence in experimental results and highlight
Fig. 14. Final distribution of matrix cracking, event b in Fig. 4.
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Fig. 15. Final distribution of ®bre±matrix shear failure, event b in Fig. 4.
areas for structural improvement without the need to necessarily produce and test costly specimens. The progressive damage model allows for variations of particular material properties and geometric differences to be investigated. Importantly, the introduction of a viscoelastic layer within the joint can be assessed on comparison with previous work on typical in-service tee joints, the con®guration of which is demonstrated by Fig. 18 [23,26,27]. The design of a typical in-service joint is different from the design of the joint described herein in both geometry and material properties of the insert material. Fig. 18 illustrates the differences in geometry of a typical standard joint with a viscoelastic joint (Fig. 1 stem from the design of the overlaminate to ¯ange bond. In the standard design, the overlaminate to ¯ange connection is much narrower than the joint with the viscoelastic insert. Furthermore, there is a raised overlaminate region in the visco-
elastic joint, which enables the viscoelastic joint to accommodate more insert resin. This feature does not exist in the standard joint. Fig. 3 compares the modulus curves of the viscoelastic insert resin with the standard crestomer insert resin. The difference in material properties of the insert material described in Fig. 3 shows that the standard crestomer resin is much stiffer than the viscoelastic material. Fig. 19 compares the load±displacement response for the viscoelastic joint and the standard joint under the same three-point load condition. For the same specimen width, the global stiffness of the standard joint is comparable with the global stiffness of the tee joint with the viscoelastic insert despite the large difference in insert material stiffness. Fig. 19 also shows that under quasi-static loading, the energy absorbed by the viscoelastic tee joint to total failure is much less than that absorbed by the standard in-service joint.
Fig. 16. Distribution of matrix cracking (red) at 6.0 mm de¯ection.
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Fig. 17. Distribution of ®bre/matrix shear failure (red) at 6.0 mm de¯ection.
Fig. 18. Typical standard in-service tee joint dimensions (mm).
Progressive damage analysis on the standard tee joint [27] describes a similar failure mechanism observed in the viscoelastic specimen. High stress concentrations are still exhibited in the overlaminate radius but again the failure occurs in the bond regions between the overlaminate and the web and/or ¯ange. Previous work on standard tee joints for example, Phillips [23] and Elliot [26], show that the failure mode is in the form of delamination in the overlaminate radius where the high stress concentrations exist and not sub-structural debonding. This discrepancy can be explained by the resin infusion process, which has been used in the more recent specimens [27] and produces much higher FRP strength limits (ILTS and ILSS). The progressive damage analysis of the viscoelastic inserted tee joint agrees favourably with experimental results, not just in gross behaviour but in the mechanisms
Fig. 19. Comparison between load±displacement curves of a standard in-service tee joint (red) and a viscoelastic inserted tee joint (blue). Both curves relate experimental tests [28].
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that cause that behaviour. The ability to assess load transfer mechanisms and failure patterns successfully allows the structural design to be varied and the subsequent response investigated without necessarily producing costly specimens. The consequence of adding a viscoelastic material into a joint that experiences quasi-static structural loading highlights the need to investigate alternative joint geometric design and production to make full use of the advantages an energy absorbing material provides in terms of attenuation of noise and vibration. 6. Conclusions A progressive damage methodology has been presented in this paper and applied with success to an example of a speci®c tee joint undergoing a three-point bend. In particular, the novel aspects of the tee joint design were assessed, namely the in¯uence of the viscoelastic insert material and the geometric variations upon the structural response to quasi-static loading. Importantly the cause and effect of failure within the joint was identi®ed and provided insight into the global response of the structure that correlated well with experimental observations and highlighting areas requiring further structural design.
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
References [1] Godwin EW, Matthews FL. A review of the strength of joints in ®bre reinforced plastics: Part 1. Mechanically fastened joints. Composites 1980;11(3):155. [2] Matthews FL, Kilty PF, Godwin EW. A review of the strength of joints in ®bre reinforced plastics: Part 2. Adhesively bonded joints. Composites 1982;13(1):29. [3] Smith CS. Structural problems in the design of GRP ships. Proc Symp GRP Ship Construction. RINA, London, October 1972. p. 33±56. [4] Gillespie JW, Pipes RB. Behaviour of integral composite joints Ð ®nite element and experimental evaluation. J Compos Mater 1978;12:408. [5] Shenoi RA, Hawkins GL. In¯uence of material and geometry variations on the behaviour of bonded tee connections in FRP ships. Composites 1992;23:335±45. [6] Shenoi RA, Violette FLM. A study of structural composite tee joints in small boats. J Compos Mater 1990;24:644. [7] Theotokoglou EE, Moan T. Experimental and numerical study of composite T-joints. J Compos Mater 1996;30:190±209. [8] Hawkins GL, Holness JW, Dodkins AR, Shenoi RA. The strength of bonded tee joints in FRP ahips. Plasti, Rubber, Compos Process Appl 1993;19:279. [9] Dulieu-Smith JM, Shenoi RA, Read PJCL, Quinn S, Moy SSJ.
[21] [22] [23] [24] [25] [26] [27]
[28]
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Thermoelastic stress analysis of a GRP tee joint. J Appl Compos Mater 1997;4:283. Shenoi RA, Read PJCL, Hawkins GL. Fatigue failure mechanisms in ®bre-reinforced plastic laminated tee joints. Int J Fatigue 1995;17:415±26. Clark JL, editor. Structural design of polymer composites Ð EUROCOMP design code and handbook London: E & FN Spon, 1996. House JR. Energy absorbing joints and their application to noise reduced sonar domes. Proc ICCM-11. Gold Coast, Australia, vol. VI, July 1997. p. 74±83. Chang FK, Chang KY. A progressive damage model for laminated composites containing stress concentrations. J Compos Mater 1987;21:834±55. Chang FK, Lessard LB. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings. Part I: Analysis. J Compos Mater 1991;25:2±43. Lessard LB, Chang FK. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings. Part II. Experiment. J Compos Mater 1991;25:44±64. Shahid I, Chang FK. An accumulative damage model for tensile and shear failures of laminated composites plates. J Compos Mater 1995;29:926±81. Sleight DW, Knight NF, Wang JT. Evaluation of a progressive failure analysis methodology for laminated composite structures. AIAA paper 97-1187, 1997. Reddy YSN, Reddy JN. An accurate prediction of failures in composite laminates using a layer-wise model. Proc Int Conf on Compos Mater, ICCM-9, vol. 3, 1993. p. 15±22. Kam TY, Sher HF. Non-linear and ®rst ply failure analyses of laminated composite cross-ply plates. J Compos Mater 1995;29: 463±82. Echaabi J, Trochu F, Pham XT, Ouellet M. Theoretical and experimental investigation of failure and damage progression of graphite± epoxy composites in ¯exural bending test. J Reinf Plast Compos 1996;15:740±55. Tolson S, Zabaras N. Finite element analysis of progressive failure in laminated composite plates. Comput Struct 1991;38:361±76. Padhi GS, Shenoi RA, Moy SSJ, Hawkins GL. Progressive failure and ultimate collapse of laminated composite plates in bending. Compos Struct 1998;40:277±91. Phillips HJ. Assessment of damage tolerance levels in FRP ship structures. PhD thesis, 1997. Junhou P, Shenoi RA. Examination of key aspects de®ning the performance characteristics of out-of-plane joints in marine structures. Compos: Part A 1996;27A:89±103. Blake JIR, Shenoi RA, House J, Turton T. Strength modelling in stiffened FRP structures with viscoelastic inserts. Submitted for publication. Elliot DM, Mechanical testing of composite joints Ð interim report, April 1994. DRA/AW/AWS/TR94212. Blake JIR, Shenoi RA, Price WG, Padhi GS. Structural response of tee joints with a viscoelastic or standard insert material undergoing a three-point bend. Ship Science Report No. 105. University of Southampton, January 1999. Turton T. Results of experimental three-point bend tests on two types of tee joint and top hat stiffener. Personal communication, November 1999.
www.elsevier.com/locate/compositesa
Progressive damage analysis of tee joints with viscoelastic inserts J.I.R. Blake a, R.A. Shenoi a,*, J. House b, T. Turton b a
Fluid Structure Interactions Research Group, School of Engineering Science, University of Southampton, High®eld, Southampton, SO17 IBJ, UK b Mechanical Sciences Sector, DERA, Farnborough, Surrey, GU14 OLX, UK Received 14 April 2000; revised 8 September 2000; accepted 11 October 2000
Abstract The purpose of this paper is to investigate the static structural response of a new type of composite tee joint containing a viscoelastic insert. The introduction of this material has proven bene®ts in terms of noise and vibration attenuation across the joint. The effects of introducing this new material on the structural response of the joint are numerically examined by using a progressive damage model. Application of this method allows the initiation and progression of failure and ultimate failure load to be predicted. Experimental results show good qualitative and quantitative agreement with the predictive damage model. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: B. Stress transfer; B. Delamination; C. Finite element analysis; D. Mechanical testing; Failure criteria
1. Introduction Large ®bre-reinforced plastic (FRP) structures, by necessity of design and production constraints, have a number of in-plane and out-of-plane joints in their topologies. In-plane joints have been the subject of attention of many researchers, for example, Godwin and Matthews [1] and Matthews et al. [2], with attention being focused on analytical treatments, numerical analyses, experimental studies, failure criteria and material aspects. Arguably, the more dif®cult problem is the one that pertains to load transfer between two orthogonally placed members meeting at a joint. The joint, an example of which is illustrated in Fig. 1, is formed by placing laminated strips of reinforcement cloth on both sides of the joint (boundary angle or overlaminate). The reinforcements are usually alternate layers of woven roving (WR) and chopped strand mat (CSM). The numbers of plies or laminae will depend on the required stiffness for the joint. The resulting gap formed between the cloth and plates is ®lled with an appropriate ®llet resin, i.e. one that is compatible with the cloth material and is generally one with a high yield strength. The weakness in this case is due to the lack of reinforcement across the connected surfaces and through the occurrence of stress concentrations associated with joint geometry and production considerations. * Corresponding author. Tel.: 144-1703-592316; fax: 144-1703593299. E-mail address: [email protected] (R.A. Shenoi).
There is now a growing body of literature on the behaviour of laminated tee joints. Early work concerned theoretical modelling using simple approaches with plane strain [3] and plane stress [4]; these corresponded to marine and aircraft applications, respectively. These efforts were rather restrictive owing to the relatively immature ®nite element analysis (FEA) capabilities at that time. More recent effort, incorporating layered (®nite) elements and a variety of failure criteria, has successfully characterised through numerical analyses both single skin [5] and sandwich [6] tee-joints under representative static loadings. This work has been supplemented by experimental programmes studying failure mechanisms [8] and stress patterns [9]. The work has been extended to studying the longterm effects of such joints under repetitive, cyclic loadings [10] including inception and progression of failure as well as approaches to life modelling. All such work has formed the basis of writing a procedure for the synthesis of design variables for typical single skin tee joints in ship and civil construction [11]. What is now being envisaged is to better utilise the joint in a dynamic response mode. One way to achieve this is to investigate the in¯uence of high strain-to-failure, viscoelastic materials. It has been shown that the use of such materials could lead to better energy absorption capabilities in structures. Preliminary work [12] has demonstrated that a tapered viscoelastic layer placed between a GRP beam and a steel supporting substrate can produce a signi®cant absorption of vibrational energy. The purpose of this paper is to consider the effects of
1359-835X/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 1359-835 X(00)00 158-5
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Fig. 1. Basic constituents of viscoelastic inserted tee joint and dimensions (mm).
placing such a material in a tee joint on its strength and stiffness characteristics.
that the use of a viscoelastic wedge absorbs compression wave energy, i.e. little energy was re¯ected at the joint interface and little was transmitted across. Furthermore, the radiated noise characteristics were much lower than those of conventional butt technology. Whereas the dynamic behaviour of the material and structure have been and are being assessed elsewhere, there is currently no knowledge of the strength and load-bearing capability of structures incorporating such materials. Furthermore, although coupon tests can and do indicate innate strength of the material itself, there is a weakness related to the manner in which it behaves within a structure. It is important therefore to understand load transfer, damage initiation and progressive build-up of damage in generic structures, such as tee joints, incorporating these viscoelastic materials.
3. Progressive damage model 2. Concepts in energy absorbent joints
3.1. Background
There are many instances in structural applications where noise and vibration transmission is unwelcome. One such example can be seen in sonar domes. Once installed on an operational boat, hydrodynamic ¯ow and supporting structural induced vibrations cause the dome to vibrate thus radiating noise and interfering with sonar sensor response. The introduction of a composite viscoelastic joint produces a vibration sink that can absorb ¯ow-generated and structure-borne noise within the dome [12]. In conventional FRP, the resins tend to have a relatively high modulus with as high a glass transition temperature Tg as possible. When matrix resins of this type are used with ®bre reinforcements, the result is a composite with high speci®c stiffness but relatively low intrinsic damping. Indeed all efforts to seek matrix resins with higher Tg (particularly in the aircraft industry) only exacerbate the problem, resulting in composites with lower and lower damping. This paper is aimed at considering the use of resin matrices with high intrinsic damping and damage tolerance, which can be optimised for speci®c frequencies and temperatures. Such composites may ®nd applications in the fabrication of new and novel structural connection concepts. House [12] describes previous work on acoustic interaction at a simple joint. It is shown that high vibration re¯ection losses can be obtained at a GRP±steel interface provided the joint input impedance is matched. This is generally coupled with a low transmission loss but at the expense of a good re¯ection loss. In all these cases, vibration energy is not lost or absorbed but merely redirected or altered by mode conversion. An alternative approach is to provide a joint that actually absorbs the energy. Current research shows that the introduction of a vibrational sink such as a viscoelastic polymeric interlayer between the GRP and steel signi®cantly attenuates the transmissibility of noise and vibration. House [12] concluded
Structural response of composite structures with intrinsic ¯exibility and non-linear material stiffness characteristics requires a non-linear ®nite element (FE) analysis. A linear analysis would assume that, despite the material failure at a location inside the joint, the global properties of the joint are not compromised in any way and the joint can be subjected to further increasing loads. In reality, there are many mechanisms for failure within the composite joint, that whilst not necessarily leading to ultimate failure at a given load, will affect the joint characteristics as the load is increased. There has been much experimental and numerical work on the failure behaviour of composite laminates. However, these investigations have concentrated on in-plane load conditions of tension, compression and shear [13±17]. Out-of-plane loading of laminated plates has received much less attention due to the material and geometric non-linearites but some important investigations can be found in Reddy and Reddy [18], Kam and Sher [19], Echaabi et al. [20], Tolson and Zabaras [21] and Padhi et al. [22]. The progressive damage of out-of-plane composite structures other than laminated plates has not been investigated in an entirely rigorous manner. Phillips [23] investigated the progressive damage of tee joints and top-hat stiffeners by considering the manual insertion of delaminations within the structure. At each load step, a visual inspection of the stress distributions determined the applicability and subsequent location of an inserted delamination. This work produced an insight into the failure mechanism of the two types of structure but was not accurate in simulating the structural response. Padhi et al. [22] successfully used FE modelling incorporating a progressive damage subroutine. This automated progressive damage modelling ®rst determines the load and
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Table 1 Material properties for the polyester/woven roving glass laminate in the ¯ange, web and overlaminate in the tee joint
single tensor polynomial failure criterion such as that proposed by Tsai. Failure is assumed to occur if the following condition is satis®ed:
Property
Value
Fi s i 1 Fij s i s j 1 Fijk s i s j s k 1 ¼ $ 1
Ex Ey Gxy Density Poisson ratio
17 100 MPa 7400 MPa 4400 MPa 1650 kg m 23 0.15
The two-dimensional form of the above polynomial is expressed as:
location at which a material in the structure ®rst fails. From that stage onwards, the failure at that location is included in subsequent analyses. The progression of damage is therefore determinable and so too the ultimate collapse load. Progressive failure analysis is based on the assumption that the damaged material can be substituted with an equivalent material with degraded properties. This can be accomplished in two principal ways: 1. Total discount method: The stiffness and strength of a failed material ply is reduced to zero. This approach may lead to an underestimation of the laminate strength because it does not recognise that ply failure is localised and that the remaining stiffness of the laminate ply is not necessarily zero. 2. Limited discount method: The reduction of stiffness depends upon the failure mode in action. For ®bre failure, the longitudinal stiffness is degraded, whereas for matrix failure, zero stiffness and strength are assigned to the failed ply for the transverse mode. The actual method used is the limited discount or stiffness reduction approach and is based on the work of Chang and Chang [13]. The method for stiffness reduction is simple but effective. For matrix cracking at a material integration point, the transverse modulus Ey and Poisson's ratio n yx are reduced to zero. However, the longitudinal modulus Ex and the shear modulus Gxy remain unchanged. When ®bre±matrix shearing is predicted at a material point, the transverse modulus Gxy and Poisson's ratio n yx are reduced to zero. However, the longitudinal modulus Ex and transverse modulus Ey remain unchanged. If ®bre failure is detected, then the material is deemed to have lost complete stiffness at the integration point.
1
F1 s 1 1 F2 s 2 1 2F12 s 1 s 2 1 F11 s 12 1 F22 s 22 1 F66 s 62 $ 1 2 In the expressions, the notations, s 1, s 2, s 6 s 6 s 12 are the in-plane stresses in the material coordinate directions. The Fij terms are the failure indices and are weighted according to the importance of individual stress components. There are different methods for deriving the failure indices; in this paper only the Tsai±Hill failure criterion is used. The failure indices for the Tsai±Hill criterion are F1 0; F11
1 2; X
F2 0; F22
F12 2
1 2; Y
1 ; 2X 2
F66
1 2 SC
3
where if s 1 . 0 then X XT, otherwise X XC and if s 2 . 0 then Y YT, otherwise Y YC.XT, XC, YT, YC and SC are strength parameters (ultimate tensile strength and ultimate compressive strength in the X, Y and XY directions). 3.3. Progressive damage assessment For the interactive polynomial criterion, if failure occurs the following expressions are used to determine the failure mode: H1 F1 s 1 1 F11 s 12 H6 F66 s 62
H2 F2 s 2 1 F22 s 22
4
The largest Hi term is selected as the dominant failure mode and the corresponding modulus is reduced to zero. Thus, H1 corresponds to ®bre failure, H2 corresponds to matrix cracking and H6 corresponds to ®bre±matrix shearing failure. The stiffness reduction method is applied and depending upon the mode of failure, the material properties are degraded accordingly.
3.2. Polynomial failure criteria
4. Numerical modelling
The next question is how to predict failure? In general, failure criteria can be categorised in two classes: independent and interactive (or polynomial). The former is simple to apply and gives the mode of failure, but it neglects the effect of stress interactions in the failure mechanism. The latter includes stress interactions in the failure mechanism, but it does not give the mode of failure. Most failure criteria for composite materials can be expressed in terms of a
4.1. Modelling basis The numerical model has been constructed and analysed using abaqus 5.8. The model is two-dimensional and made up of 3691 four-noded CPS4 elements. A plane stress analysis is used with specimen thickness of 150 mm. The joint geometry is given in Fig. 1 and the material properties in Table 1.
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Fig. 2. Adopted mesh for ¯exible tee joint (detail of overlaminate radius and web to overlaminate bond shown in inset). Viscoelastic insert mesh not shown. Loading conditions also shown.
A large number of ®nite element models have been constructed and tested in order to optimise the ®nal progressive damage model. The progressive damage model in essence requires the stiffness degradation of localised points within the structure to represent damage. The less re®ned the ®nite element mesh, the more signi®cant the effect of localised stiffness reduction on the global strength of the joint. Therefore while a coarse mesh reduces computational expense, its adverse effect on the accuracy of the progressive damage model may be signi®cant. The ®nal adopted model thus re¯ected these two opposing considerations. Previous experimental and numerical studies (e.g. [5,23] showed that large through-thickness stresses are expected in
Fig. 3. Stress±strain curves comparing relative stiffness of the standard crestomer inset material used in in-service tee joints and the viscoelastic insert material.
the overlaminate prior to delamination. The density of elements is increased in the overlaminate, focussing the attention of the structural response and aiming to improve the accuracy of the progressive damage model. Each ply of the overlaminate is represented by one element of thickness 0.61 mm. Fig. 2 shows the adopted mesh layout. 4.2. Loads, material properties and boundary conditions Fig. 2 describes the boundary conditions applied in the numerical models. The numerical model is constrained at nodes on the upper surface of the ¯ange, 500 mm apart. Both nodes are prevented from translation in the y-direction (plane of the web) but to restrict a rigid body motion, one of the nodes is restricted in the x-direction also (in the plane of the ¯ange). The progressive damage model increases the displacement from 0 to 10 mm in steps of 0.1 mm. The displacement rate is considered small and so the response of the joint is quasi-static. Constantly increasing displacement as opposed to increasing load is used in order to attempt to pick up any stress relief. Linear material properties have been used for the overlaminate, the web and the ¯ange. The viscoelastic insert exhibits nonlinear stress±strain characteristics and is modelled herein using a bilinear stress±strain curve illustrated in Fig. 3. This particular viscoelastic material was produced and tested at DERA Farnborough and is custom made for noise attenuation at speci®c frequencies and temperatures. The abscissa is foreshortened for clarity of the initial stiffness of this particular elastomer, but in®nite strain is considered for numerical analysis.
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4.3. Incorporation of progressive damage model
5. Validation and results
The procedure for progressive failure analysis adapted in this context is given below. This is programmed into a fortran standard user subroutine USDFLD, incorporated and used in the abaqus 5.8 processor. This routine allows the user to de®ne material properties as functions of the ®eld variables at a material point. The material properties of the joint were de®ned to be dependent upon three ®eld variables. The ®rst ®eld variable was the contribution of s 1 (H1) towards the failure index. The second and third ®eld variables were the contributions of s 2 and s 12 (H2 and H6, respectively) to the failure index:
5.1. Experimental results
1. At the load step, a geometric non-linear analysis is performed until a converged solution is obtained. 2. The stresses at every integration point are evaluated. 3. The failure polynomial is determined and if larger than 1.0.¼ 4. ¼the contribution of each stress component towards the failure index (Hi) is computed and the stress component, which contributes the maximum is identi®ed. 5. Depending on the largest Hi term, the failure mode is determined and the corresponding material properties are degraded accordingly. 6. A geometrically non-linear analysis is again performed for the degraded structure to re-establish equilibrium. If no more failure is detected then¼ 7. ¼the next load step is applied, 8. ¼otherwise the structure is degraded again according to steps 3±6. At any load step, a ply failure load is the load at which the failure index (Fis i 1 Fijs is j) reaches a value of unity at any material integration point. At some point in the analysis, the global stiffness becomes zero or negative and this is taken as the inability of the structure to support additional load. This location is identi®ed as the ultimate failure load. Global failure can also be interpreted based upon the degree of failure, mode and location that exists. This interpretation is important if there is likely to be considerable residual strength of a restrained substructure that remains loaded. For example, joint debonding does not lead to full structural failure as there still exists residual stiffness from the ¯ange, however the joint itself is now considered useless. The numerical models are based upon continuum mechanics and as such do not consider the discontinuity produced by physical debonding and the related redistribution of stress. Therefore, whilst the predicted model response shows the existence of positive global stiffness, in real terms the predicted amount of material failure in speci®c regions would have produced complete redundancy of the physical structure. This method of interpreting speci®c failure to cause structural redundancy was employed in the experimental tests.
645
The numerical model is validated using the results of the experiments upon the physical joint. Due to the scarcity of these unique joints and the cost of producing the viscoelastic insert material and manufacturing the joints, only one destructive experiment was performed. The physical specimens were fabricated in Vosper Thornycroft shipyard. The web and ¯ange were cut from a single laminated plate produced using a resin infusion process. The viscoelastic insert was bonded on top of the ¯ange underneath the web. The cured surfaces of the web and ¯ange were abraded before the overlaminate was laid up either side of the joint. The overlaminate material was then resin infused. The specimens were loaded in a standard three-point bend described in Fig. 2. This type of loading is considered to be representative of an UNDEX 1 shock event experienced in marine environments [24]. The hull plating either side of the stiff joint is subjected to large negative pressures caused by shock bubble implosion. Rollers with a 30-mm diameter were used at the contact points on the specimen, 500 mm apart. A displacement was applied at the structure's centreline and the reaction load at the point of application was measured against increasing displacement. The tests were recorded on video so that the failure events from the video could be related to the load against displacement curve. Fig. 4 describes the experimental results of load versus displacement and shows a series of load drop-offs when substantive failure leads to progressive increase in structural ¯exibility. The events corresponding to the various load drops are listed in Table 2. Fig. 5 illustrates the ®nal failure of the joint and position of initial failure. The initial response of the joint is fairly linear upto x (Fig. 4 although there appears to be some stiffening between 2 and 3 mm de¯ection. Correlation of the response curve with the video record shows that at a reaction load of 7 kN, location x, a crack has initiated in the bond between the overlaminate and the web at the base of the web, shown by the annotation in Fig. 5. This cracking accounts for the load drop at x. Between x and y, the joint response is again linear in nature. From the video record, the crack propagates up the overlaminate to web bond until the overlaminate detaches completely from the web at y. This marks the point of structural redundancy, and therefore ultimate failure, even though the joint can take additional load through the deformation of the ¯ange. 5.2. Validation of global structural response Comparisons between numerical and experimental results are made with regard to load versus displacement curve and the initiation and progression of failure and 1
Underwater explosion.
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Fig. 4. Comparison of experimental and predicted load±displacement results. The letters correspond to failure events shown in Table 2.
ultimate failure load. Numerical and experimental results of load against displacement are shown in Fig. 4. Fig. 4 and Table 2, which correlates the failure events in Fig. 4 with physical observations, demonstrate that the predicted response of the tee joint agrees favourably with the experimental results. Initial stiffness of the numerical model is comparable to the experimental result. The predicted stiffness up to the large stress relief at event a is fairly linear. However, unlike the experimental results, which exhibit a small amount of stiffening between 2 and 3 mm de¯ection, there appears to be a slight increase in joint ¯exibility and originating at around 0.5 mm. The large load drop at a is over-predicted in terms of joint de¯ection, but the reaction load at which this drop occurs agrees well. The trend of the numerically computed results of load versus de¯ection between a and b compares favourably with the experimental results. However the ®nal failure load and de¯ection is, at ®rst sight, in poor agreement with experiment.
5.3. Internal load transfer and failure patterns To understand the global structural response predicted by the numerical models, it is necessary to investigate the causal factors namely internal load transfer mechanisms and failure patterns. The slight departure of the initial response of the joint from linear behaviour, originating at 0.5-mm de¯ection, can be explained in Fig. 6, which describes the amount of matrix cracking within the structure at this de¯ection. It is evident that at this very low de¯ection of 0.5 mm, there is already a small amount of matrix cracking occurring within the overlaminate to web bond, illustrated by the proportion of red colour in this region. The asymmetry in the distribution of damage is thought to arise from numerical rounding within the mesh generation. Any asymmetry of the generated nodal coordinates in the yz-plane (Fig. 2 Ð z-direction is out of the paper) would lead to asymmetric loading and therefore predicted damage.
Fig. 5. Final failure of viscoelastic inserted joint, event y in Fig. 4.
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Fig. 6. Distribution of damage (red) in terms of matrix cracking at a de¯ection of 0.5 mm.
Fig. 7 demonstrates the through-thickness stress distribution and Fig. 8 the shear stress distribution within the FRP of the joint at 0.5 mm de¯ection. Increasing the de¯ection from 0.4 to 0.5 mm results in an increase in stress magnitude greater than that observed in prior increments. The distribution of through-thickness stress and shear stress remains fairly constant, with higher stress gradients at the outer plies of the overlaminate radius. The deviation from the linear behaviour of the numerical model's response in the ®rst few increments corresponds to the inception of matrix cracking in the overlaminate to web bond demonstrated in Fig. 6. The load at which the large stress relief occurs in both the numerical model (point a) and in the experimental model (point x) agree well. However, the numerical model over predicts the de¯ection at which this event occurs. The reason for the stress relief can be seen in Figs. 9 and 10, which shows the distribution of matrix cracking prior to (Fig. 9 and after (Fig. 10 the event. The distribution of damage is represented by the red
colour in the above ®gures. Prior to event a, there is already a considerable amount of matrix cracking within the overlaminate to web bond, but the largest rate of damage occurrence happens at the de¯ection corresponding to a reaction load of 6.78 kN. This sudden increase in damage may be analogous to the inception of a crack in the base of the overlaminate to web bond seen experimentally. In the region x to y in Fig. 4, the crack between the web and the overlaminate in the physical specimen propagated up the web away from the ¯ange leading to complete debonding at y. The numerical models predict a different failure pattern between the equivalent points a and b. Figs. 11±13 show the distribution of through-thickness stress and shear stress in the FRP and the proportion of matrix cracking predicted at a de¯ection of 4.9 mm. Although there is little change in the distribution of stresses and negligible increase in shear stress magnitude in the interval of 4.8±4.9 mm, the through-thickness stress increases markedly from a de¯ection of 4.8±4.9 mm over stress increases in previous increments. This sudden increase coincides with the initiation
Fig. 7. Distribution of through-thickness stress in the FRP at 0.5 mm.
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Fig. 8. Distribution of shear stress in the FRP at 0.5 mm.
and propagation of matrix cracking through the outer plies of the overlaminate radius illustrated in Fig. 13. The through-thickness stress concentrations, demonstrated in Fig. 11, are on the right-hand side of the joint and yet Fig. 13 describes the initiation and propagation of matrix failure on the left-hand side. The largest stress contributor to the failure polynomial determines the dominant failure mode. However, due to the interactive nature of the stress contributions upon the failure polynomial, the in-plane stress contribution (which is much greater on the left-hand side of the joint) affects the location of predicted matrix failure. Absolute shear stress values are comparable either side of the joint.
The de¯ections of 4.8±4.9 mm correspond to the minima in the predicted load±de¯ection graph (Fig. 4. After this minima, the incidence of matrix cracking increases throughout the overlaminate radius accompanied by ®bre±matrix shear failure whilst the incidence of failure within the web to overlaminate bond region is reduced. Consequently, the ®nal failure load and mode predicted by the numerical model is in disagreement with the experimental results with Figs. 14 and 15 describing the predicted ®nal distribution of damage. However, observation of the failure pattern within the predicted model leads to conclusions for the point of structural redundancy. For example, Figs. 16 and 17 show that there is total matrix and shear failure throughout the
Fig. 9. Distribution of damage (red) in terms of matrix cracking prior to event a in Fig. 4.
Fig. 10. Distribution of damage (red) in terms of matrix cracking after event a in Fig. 4.
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Fig. 11. Distribution of through-thickness stress in the FRP at 4.9 mm.
bond region, in real terms most likely describing detachment of the overlaminate from the web. This occurs at a de¯ection of 6 mm, which corresponds to a reaction load of 7 kN, which is much lower than the predicted ®nal failure load and in closer agreement with experimental observations. 5.4. Discussion The progressive damage model has provided good qualitative and quantitative agreement with experimental results. However, it is important to treat the results with caution given the limited number of experiments that could be performed. The conclusions drawn are based on the supposition that the experimental data is representative of this speci®c joint design incorporating these materials and under the described form of loading. The con®dence in
the numerical modelling is re¯ected by the agreement with the available experimental data for these speci®c joints (bearing in mind the above supposition) and between the progressive damage methodology and a set of experimental tests, carried out simultaneously with the tee joints, on top hat stiffener sections [25]. To improve the modelling further, the material strength limits require re®ned de®nition. Phillips [23] demonstrated the importance of tolerance speci®cations for published material properties on numerical modelling. Furthermore, production processes rarely result in two specimens ever being the same, thereby providing some uncertainty as to which material strength limits to employ. This uncertainty existed in the numerical modelling of the specimen described herein. As described in Section 4.1, the adopted ®nite element mesh is a compromise between computational expense and
Fig. 12. Distribution of shear stress in the FRP at 4.9 mm.
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Fig. 13. Distribution of matrix cracking (red) at 4.9 mm de¯ection.
accuracy. The number of elements was increased in the progressive damage model until a state of convergence in terms of load±displacement result was reached. The only effect of increasing the number of elements past this point was to increase the amount of CPU time for solution. Therefore, to increase correlation between experiment and theory in the initial location of failure, the numerical strength limits for interlaminar tensile strength (ILTS) and shear strength (ILSS) were reduced in the bonded regions. Table 3 describes the assumed strengths used in the progressive damage model throughout the FRP material and the reduction of these limits in the overlaminate to web and overlaminate to ¯ange bond regions. The necessity for strength reduction in these regions highlights a possible structural de®ciency caused in the production of the specimens through lack of resin infusion. Observation of the failed physical specimen re¯ected this
Table 3 Comparison between assumed material properties through the joint (`Bond' failure limits relate to web-to and ¯ange-to overlaminate failure limits. `FRP' relates to strength limits applied throughout the rest of the FRP material) Failure limits (MPa) In-plane
ILTS
Shear
Insert
FRP
Bond
FRP
Bond
FRP
Bond
261
261
31.5
7.5
50
20
4.65
hypothesis; there was little ®bre breakage in the bonded area and a clean glassy surface remained on each of the join faces. Consequently, the progressive damage model can reinforce con®dence in experimental results and highlight
Fig. 14. Final distribution of matrix cracking, event b in Fig. 4.
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651
Fig. 15. Final distribution of ®bre±matrix shear failure, event b in Fig. 4.
areas for structural improvement without the need to necessarily produce and test costly specimens. The progressive damage model allows for variations of particular material properties and geometric differences to be investigated. Importantly, the introduction of a viscoelastic layer within the joint can be assessed on comparison with previous work on typical in-service tee joints, the con®guration of which is demonstrated by Fig. 18 [23,26,27]. The design of a typical in-service joint is different from the design of the joint described herein in both geometry and material properties of the insert material. Fig. 18 illustrates the differences in geometry of a typical standard joint with a viscoelastic joint (Fig. 1 stem from the design of the overlaminate to ¯ange bond. In the standard design, the overlaminate to ¯ange connection is much narrower than the joint with the viscoelastic insert. Furthermore, there is a raised overlaminate region in the visco-
elastic joint, which enables the viscoelastic joint to accommodate more insert resin. This feature does not exist in the standard joint. Fig. 3 compares the modulus curves of the viscoelastic insert resin with the standard crestomer insert resin. The difference in material properties of the insert material described in Fig. 3 shows that the standard crestomer resin is much stiffer than the viscoelastic material. Fig. 19 compares the load±displacement response for the viscoelastic joint and the standard joint under the same three-point load condition. For the same specimen width, the global stiffness of the standard joint is comparable with the global stiffness of the tee joint with the viscoelastic insert despite the large difference in insert material stiffness. Fig. 19 also shows that under quasi-static loading, the energy absorbed by the viscoelastic tee joint to total failure is much less than that absorbed by the standard in-service joint.
Fig. 16. Distribution of matrix cracking (red) at 6.0 mm de¯ection.
652
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Fig. 17. Distribution of ®bre/matrix shear failure (red) at 6.0 mm de¯ection.
Fig. 18. Typical standard in-service tee joint dimensions (mm).
Progressive damage analysis on the standard tee joint [27] describes a similar failure mechanism observed in the viscoelastic specimen. High stress concentrations are still exhibited in the overlaminate radius but again the failure occurs in the bond regions between the overlaminate and the web and/or ¯ange. Previous work on standard tee joints for example, Phillips [23] and Elliot [26], show that the failure mode is in the form of delamination in the overlaminate radius where the high stress concentrations exist and not sub-structural debonding. This discrepancy can be explained by the resin infusion process, which has been used in the more recent specimens [27] and produces much higher FRP strength limits (ILTS and ILSS). The progressive damage analysis of the viscoelastic inserted tee joint agrees favourably with experimental results, not just in gross behaviour but in the mechanisms
Fig. 19. Comparison between load±displacement curves of a standard in-service tee joint (red) and a viscoelastic inserted tee joint (blue). Both curves relate experimental tests [28].
J.I.R. Blake et al. / Composites: Part A 32 (2001) 641±653
that cause that behaviour. The ability to assess load transfer mechanisms and failure patterns successfully allows the structural design to be varied and the subsequent response investigated without necessarily producing costly specimens. The consequence of adding a viscoelastic material into a joint that experiences quasi-static structural loading highlights the need to investigate alternative joint geometric design and production to make full use of the advantages an energy absorbing material provides in terms of attenuation of noise and vibration. 6. Conclusions A progressive damage methodology has been presented in this paper and applied with success to an example of a speci®c tee joint undergoing a three-point bend. In particular, the novel aspects of the tee joint design were assessed, namely the in¯uence of the viscoelastic insert material and the geometric variations upon the structural response to quasi-static loading. Importantly the cause and effect of failure within the joint was identi®ed and provided insight into the global response of the structure that correlated well with experimental observations and highlighting areas requiring further structural design.
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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