Modelling the fatigue behaviour of sandwich beams under monotonic, 2-step and block-loading regimes

Composites Science and Technology 59 (1999) 471±486

Modelling the fatigue behaviour of sandwich beams under monotonic, 2-step and block-loading regimes S.D. Clark a, R.A. Shenoi a,*, H.G. Allen b a

Department of Ship Science, University of Southampton, High®eld, Southampton S017 1BJ, UK b Department of Civil Engineering, University of Southampton, Southampton, UK

Received 4 September 1997; received in revised form 23 March 1998; accepted 20 April 1998

Abstract FRP composite sandwich materials are being increasingly used in high-performance structural applications because of their high sti€ness and low weight characteristics. At present, relatively little is known about the long-term performance of such structures, especially under multi-step loading conditions. The aim of this paper is to address such fatigue behaviour by using a fatigue model veri®ed by experimentation. The fatigue model is based on the fatigue modulus concept (degradation of sti€ness) which is proposed for core-dominated behaviour. Two non-linear cumulative damage models derived from the chosen sti€ness degradation equation are examined in context with the linear Miner's damage summation and compared with available experimental data for 2-step loading. The models are then applied to examine the in¯uence of 2-step block loading. Finally, conclusions are drawn regarding the aptness of the chosen models. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Structural materials; B. Fatigue; C. Sandwich; Sti€ness degradation

1. Introduction Composite materials are used in a variety of structural applications. One combination which can be used as an ecient structural component for supporting axial and/or ¯exural loads is a sandwich beam/panel which consists of a relatively low-density core sandwiched between two sti€ skin materials. This combination produces a relatively strong lightweight beam which is often used for high-performance applications where minimal structural weight and maximum sti€ness/strength is important. Currently, relatively little is known about the longterm performance of such structures under both single load and spectrum loading conditions. This is particularly so with regard to the fatigue behaviour of the core material. Most published works on sandwich structures have been typically experimental in nature where fatigue behaviour has been characterised by de¯ection-cycles and stress/life (S-N) trends [1]±6]. In cases where the core material has failed in shear, there seems to be some agreement that there are no signi®cant signs of core deterioration during cycling until a point close to failure * Corresponding author. Tel.: +44 1703 592316; fax: +44 1703 593299.

where beam sti€ness is suddenly lost as a result of core degradation. Only the beam/panel de¯ections show a small increase with the number of load cycles up until close to failure which is thought, though not proved, to be due to creep e€ects. The energy absorption obtained from hysteresis loop area measurements during cycling is low and constant and there are no signi®cant changes in sti€ness for most of the beam's life. The beam's response is therefore sometimes assumed to be linear if stress amplitudes do not exceed the static linear limits. Modelling sandwich beam behaviour has been far more limited, most approaches being based on a simple lifetime S-N approach, which does not strictly model the accumulation of fatigue damage with increasing number of cycles. As far as the e€ect of load sequence is concerned, there have been no theoretical or experimental modelling investigations on sandwich beam behaviour. The objective of this paper therefore is to introduce fatigue models for sandwich beams under both single step and multi-step loading. In order to verify the chosen models, tests under these loading conditions have been undertaken. A general characterisation of fatigue behaviour is given and stress/life (S-N) diagrams pertaining to the core shear failure mode are presented. A fatigue model is then proposed for core-dominated behaviour which is based on a sti€ness-degradation

0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd.. All rights reserved. PII: S0266 -3 538(98)00088 -8

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approach. In order to predict the e€ect of multi-step loading, three cumulative damage models are investigated. One of the cumulative damage models represents the standard linear Miner's rule whilst the other two are non-linear models, one based on the failure modulus, the other on core shear strain. The models are then compared with experimental data for 2-step loading and conclusions drawn regarding load sequence e€ects for the foam core material and the appropriate choice of cumulative damage model.

2. Fatigue modelling 2.1. General Fatigue damage in composites and other related materials has often been modelled by using the reduction in strength or sti€ness. With strength-degradation approaches, the residual strength is determined from a static test after fatigue cycling, so that a series of tests are required to determine a single strength degradation curve. The Strength Life Equal Rank Assumption (SLERA) is often assumed, which states that the rank in static strength is equal to the rank in fatigue life. This assumption appears to be valid for a wide range of ®brereinforced plastics where the scatter in fatigue data is primarily due to variations in static strength, provided that the failure mode does not change [7]. A statistical distribution of strengths determined from static tests of undamaged specimens until failure is often used to relate the static scatter to the scatter of residual strengths in fatigue [7±9]. Sti€ness degradation methods have the advantage of being able to allow measurement of e€ective sti€ness during cycling without destruction of the specimen, so that a sti€ness degradation curve can be obtained from a single test. Smaller numbers of specimens are required and average results can be used rather than the use of a full statistical analysis. However sti€ness can be de®ned in di€erent ways. Usually, it is taken as a modulus term where the reduction in sti€ness can be measured from the linear portions of stress/strain graphs at di€erent cycle numbers. Yang et al. [10] assumed that the degradation rate is a power-law function of the number of load cycles. This approach is mainly suitable for ®bredominated response where the stress/strain curve is linear. Hwang and Han [11] introduced a model based on the fatigue modulus concept, an approach particularly suitable for resin-dominated behaviour. During fatigue cycling, the stress/strain curve changes, causing a reduction of fatigue modulus. The fatigue modulus at a speci®c load cycle, n, is represented on the stress/strain curve by a line drawn from the origin to the resultant strain at the applied stress level.

2.2. Sti€ness degradation approach 2.2.1. The fatigue modulus concept In this approach, the reduction in sti€ness is assumed to be proportional to the increase in observed de¯ection with number of cycles. As a starting point, an approach based on the fatigue modulus concept by Hwang and Han [11] will be used. This has shown to be suited to matrix-dominated laminate behaviour where shear is the primary deformation mechanism. The model is therefore most suitable for describing the fatigue behaviour of polymer foams. The term `fatigue modulus' is the ratio between the applied stress and the resultant strain at a given number of cycles and is shown schematically in Fig. 1. As the number of cycles increases at an applied shear stress,  a, the resultant shear strain, (n), at cycle n becomes larger until it reaches the failure shear strain, u. It is assumed that the strain criterion holds whereby failure occurs when the cycle-dependent fatigue strain is equal to the static failure strain, i.e. (Nf)= u. The fatigue modulus, Gf (n,r) at stress ratio r and cycle number n can be expressed by Gf …n; r† ˆ a = …n†:

…1†

Thus knowledge of the stress/strain plots at every cycle number is not required; measurement of the fatigue modulus can be made directly from a knowledge of the strain (de¯ection) history for the given material. 2.2.2. Degradation equation The rate of decrease of fatigue modulus as proposed by Hwang and Han [11], can be related to an empirical powerlaw function of the form AnC. The theoretical decrease in modulus from an initial static value can be expressed as Gf …n† ˆ G0 ÿ AnC ;

…2†

where Gf(n) and G0 are the transient fatigue modulus and instantaneous static modulus, respectively, and A and C are material constants to be determined from

Fig. 1. Fatigue modulus concept.

S.D. Clark et al. / Composites Science and Technology 59 (1999) 471±486

experimental data. This is achieved by converting the experimental de¯ection response into fatigue modulus term (see Section 4.1.2). Assuming a normal distribution, the constants are determined by using least-square curve-®tting techniques. It is shown later that the fatigue process consists of an initiation period followed by a period of damage progression. The number of cycles, n, de®ning the initiation of fatigue damage is given by nif. Because fatigue damage is only observed after the initiation of fatigue damage, the model has to modi®ed because it is applied to the secondary region of the fatigue-modulus response, i.e. n5nif. Therefore the cycle number, n, must be replaced by n-nif. After closer examination of the secondary region of the experimental de¯ection response, the power-law formulation shown in Eq. (2) was deemed to be less satisfactory than an exponential function. Thus Eq. (2) was modi®ed by using an exponential function of the form AnC and is shown below: where n4nif ; Gf …n† ˆ G0 ; Gf …n† ˆ G0 ÿ Ae…ÿnif †C ; where n5nif ;

…3†

where n is the number of cycles carried out and by using the fatigue-modulus concept Gf(n)= a/ f(n) where  a the applied shear stress and f(n) the fatigue component of the resultant shear strain. 2.2.3. S-N equation A non-linear S-N curve can be derived from the nonlinear core modulus degradation equation shown in Eq. (3). By assuming the failure strain criterion, the resultant cyclic shear strain at failure is equal to the ultimate static shear strain ( f(Nf)= u). The fatigue modulus at failure therefore becomes Gf(Nf)= a/ u. This assumes a linear stress/strain curve. An applied stress ratio, r can also be derived where Gf …Nf † a u a ˆ  ˆ ˆ r; G0

u u u

473

2.3. Cumulative damage model and multi-stress life prediction 2.3.1. General damage model Assuming constant frequency and environmental conditions, fatigue damage, D accumulates from an initial damage state, usually equal to zero, at zero cycles to a ®nal failure value, usually equal to unity. For constant loading D ˆ 0 @ n ˆ 0; D ˆ 1 @ n ˆ Nf :

…6†

For a sequence of `m' loadings: D ˆ 0P@ n ˆ 0; Dˆ m iˆ1
Di ˆ 1 @ n ˆ Nf ;

…7†

where Di is the damage experienced at load level i. The total damage is the summation of all the damage components at each loading level. Thus for 2-step loading, the residual life of a beam can be determined from the residual damage, Dr. This is represented schematically in Fig. 2. If N1 and N2 are the expected lives under the ®rst and second loads, respectively, then the remaining damage can be expressed as Dr ˆ 1 ÿ D12 ;

…8†

where D12 is the level of damage experienced under the ®rst stress level and equated to an equal amount of damage at the start of loading at the second stress level. The remaining life is therefore the number of cycles to failure under the second stress level, N2, minus the number of cycles under the second load that equates to the already accumulated damage level D12 under the ®rst load.

…4†

where r is equal to the ratio of applied stress to ultimate static stress (r= a/ u). At failure, (n=Nf) and rearranging Eq. (3), the non-linear S-N equation becomes Nf ˆ nif ‡

ln‰B…1 ÿ r†Š : C

…5†

The above equation, where B=G0/A, can be used to predict the number of cycles at failure for di€erent applied stress levels thus de®ning the ®tted curve to the S-N data. The constants A and C are determined from Eq. (2) as described in Section 2.2.2.

Fig. 2. Schematic illustration of determination of residual life: 2-step loading.

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(C) Model III: In this case the damage function is de®ned in terms of shear strain

2.3.2. Cumulative damage equations Di€erent forms of the cumulative damage parameter, D, can be chosen depending on the degree of linearity of the degradation response. Three models are investigated. The ®rst model is linear, based on `number of cycles', the second a model based on changes of `modulus' whilst the third model is based on changes of `strain'. Damage is assumed to initiate when fatigue damage is ®rst observed, i.e. at n=nif. At n=Nf the damage is equal to one. For the purpose of all the cumulative damage models investigated it is assumed that: D…n† ˆ 0; 04D…n†41;

D…n† ˆ

…9†

where n4nif ; where nif 4n4Nf :

…n ÿ nif † ; …Nf ÿ nif †

3. Experimental characterisation

where n5nif :

3.1. Material properties and construction method The chosen core material (Airex C70.130) is of thermoset cross-linked closed cellular foam construction with a density of 130 kg/m3. It is commonly used in the marine industry in hullforms and other structural details of small fast craft. The skin materials consist of hybrid glass/kevlar/epoxy construction of equal thickness giving a symmetrical cross-section about the beam's neutral axis. Details of the skin material properties as determined from individual coupon tests are shown in Table 1. Core material properties shown in Table 2 have been taken from manufacturers data since separate static tests using small core cuboid specimens displayed similar values. Sandwich panels were constructed using mixed prepreg/handlayup and vacuum bagging techniques. Firstly, the compressive skin in prepreg form was layed down and vacuum bagged using a pressure typically 80% of an atmosphere. Next, the skin was cured in an oven (typically 80 C) for approximately 10 h. The core was then bonded to the compressive skin and the

…10†

(B) Model II: The damage function is de®ned in terms of the fatigue modulus as: D…n† ˆ

G0 ÿ Gf …n† G0 ÿ Gf …Nf †

…11†

By manipulation of Eq. (3) this leads directly to D…n† ˆ

e…nÿnif †C e…Nf ÿnif †C

…13†

Using Eq. (3) and stress-strain relationships derived from Fig. 1, the damage function becomes: h r i e…nÿnif †C  …14† where n5nif D…n† ˆ 1 ÿ r B ÿ e…nÿnif †C

(A) Model I: The most basic linear cumulative damage model is that proposed by Miner [12] which was originally derived from energy considerations. It states that the amount of damage at a given number of cycles is the ratio of the current cycle number to the number of cycles to cause fatigue failure. In this case, the damage model occurs after the initiation of damage and can be expressed as: D…n† ˆ

…n† ÿ …0†

…Nf † ÿ …0†

where n5nif

…12†

Table 1 Properties of sandwich facings Skin type

Layup

Glass/Aramid & Epoxy

RE210 5QEA1200 

Thickness (mm)

Young's modulus (kN/mm2)

5.5

18.0



Tensile strength (N/mm2) 289 



Layup code: EÐE-glass, AÐaramid, RÐreinforced balanced 0 /90 woven roving, QÐbalanced quadriaxial 0 /90 /+45 /ÿ45 woven roving, No. weight of cloth per unit area (g/m2). Table 2 Properties of core materials Core type Chemical makeup

Nominal, density (kg/m3)

Thickness Young's modulus (mm) (N/mm2)

Shear modulus (N/mm2)

Shear strength (N/mm2)

C70.130 (Airex)

130

100

50.0

1.90

Rigid cross-linked PVC & Polyisocyanate (thermoset)

120.0

S.D. Clark et al. / Composites Science and Technology 59 (1999) 471±486

adjacent core layer using a ®lled epoxy resin. Typically core thickness of 50 mm were bonded at a time; greater core thicknesses were bonded in two separate operations. Curing was typically carried out at lower temperatures (45 C) because core materials were not able to withstand high temperatures. Curing took approximately 6 h at a vacuum level of 80%. The tensile skin was then laid up using a hand layup process on the completed foam core using epoxy resin. It was allowed to cure by using the same temperature as for the core for approximately 10 h at a vacuum level of 80%. The constructed panel was then sliced into beams which had a width of 200 mm and a total length of 1600 mm. 3.2. Fatigue rig Fatigue tests were conducted on a rig where the beam was loaded at eight equally spaced points along its length (using pneuamatic rams) in order to simulate a uniform load. The ends of the beam were simply supported with a span between supports of 1300 mm. Generally, the nature of the pneumatic system means that the loading rate ((maximum load)/(time to reach maximum load)) and the unloading rate are constant for a given beam. Thus a symmetrical trapezoidal wavepro®le is achieved. Measurements of the applied load and central de¯ection were taken. All tests were carried out at 23 ‹ 5 C. Further details of the apparatus are discussed in detail elsewhere [13] and hence not elaborated here. 3.3. Test results Fatigue tests were undertaken at a frequency of 0.5 Hz and an applied stress ratio (ratio min/max applied stress), R=0. The applied loads were chosen between 30% and 80% of the ultimate static load. Typically, a minimum of three beams at each of a minimum of three di€erent loading levels were chosen. For the ramp loading part of the applied trapezoidal waveform, the core shear strain rate was typically 7±10% strain per second. In some cases, residual strength tests were undertaken in 4-point bending on a separate rig because

Fig. 3. Core shear cracking fatigue failure mode: mode A.

475

the fatigue rig had not sucient load capacity to do the tests. Good correlation between the failure core shear stresses calculated using sandwich beam theory (Eq. (A.6)) was observed for the two bending con®gurations. 3.3.1. Failure mode The fatigue failure mode for the beams was brittle core shear cracking. The onset of damage during fatigue is visible, but only occurs in a short number of cycles just before ®nal failure of the beam. The location of damage in all cases was in a region close to the supports (50±80 mm) where the shear stress is a maximum and constant between the support and the adjacent load application point. In some fatigue cases, Mode A failures were observed, i.e. a clean crack from top to bottom face, at approximately 45 ; see Fig. 3, whilst in others Mode B failures were observed. In the case of Mode B failures, the crack ran from the top face, at a slightly steeper angle than for a Mode A failure, to one of the core interfaces between adjacent core layers. Numerous small cracks then propagate in a horizontal layer along the interface before propagation of a single crack again to the opposite skin aligned approximately in the same orientation as the ®rst single crack; see Fig. 4. Electron microscope photographs show that the cells appear to be completely pulverised where the crack has propagated in the region along the interface between adjacent core layers (Mode B type failure) before propagating quickly to the outside edges. This suggests that cracking in this region takes time to propagate as the crack opens and closes during cyclic loading. For Mode A failures, the fracture surface is quite smooth with cleanly broken cells indicating that fracture in this region is rapid. 3.3.2. De¯ection response For core shear failures, the beam deforms slowly at a constant rate until near failure. Close to the end of the beams life, the rate of de¯ection starts to increase rapidly in an exponential manner till failure. This region is associated with a sudden loss of strength and sti€ness. A typical de¯ection response is shown in Fig. 5.

Fig. 4. Core shear cracking fatigue failure mode: mode B.

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Fig. 5. Fatigue de¯ection versus number of cycles response.

Fig. 6. S-N curves for sandwich core failures.

3.3.3. S-N Data The fatigue data has been plotted in applied stress versus log life (S-logN) format. S-N results for other beam types of di€ering cores densities from published data tested at approximately the same frequency (0.5±2 Hz) have been added for comparison purposes; see Fig. 6. Here the data has been normalised for core density and the applied stress ratio as discussed elsewhere [14]. It is apparent that whilst there is some scatter in the experimental data, correlation with the other published data appears to be fair. The trends indicate that, in the load range considered, the experimental data points lie approximately on a line which is consistent with the approach used with other published works [1±4]. However, after 106 cycles, it is observed that the S-N data tends to depart slightly from a straight line ®t indicating

that another form of S-N equation may be more suitable in this region. 3.3.4. Multi-step loading A combination of low-high loads and then high-low loads have been carried out to investigate the in¯uence of load sequence. The beams were loaded for 50% of their average fatigue life at each respective load. Those beams that had not failed were tested for their residual static strength under static loading conditions. The results are also shown in Table 3. It is observed that there is very little loss of strength throughout most of the beam's life. Some of the results tend to indicate that there may be some slight load sequence dependency (i.e. high-low load more damaging). This is shown by two of the results for the high/low load combination showing a strength reduction of more than 25%.

Table 3 2-step loading tests; static failure load=9661 kg Beam No.

1st load (N/mm)

2nd load (N/mm)

1st cycles

2nd cycles

%1st %2nd Life Life

% Total life

Residual strength (kg)

% Static load

Failure modes

S01 a S02 a S03 a S04 a S05 a S06 a S07 a S08 a S09 a S17 c S18 c S19 c S20 c S21 c S22 c S23 c S28 c

28.21 27.75 28.49 34.37 35.39 34.98 27.75 28.61 28.64 35.45 32.02 40.88 42.89 42.21 42.41 41.85 35.3

35.48 35.74 35.01 41.83 42.22 42.68 41.75 42.73 42.47 27.9 28.15 34.76 34.89 29.33 28.55 28.86 28.05

150 000 150 000 150 000 20 252 20 255 20 252 150 000 150 057 150 000 40 000 40 000 1500 1500 1578 1580 1527 40 000

20 000 20 000 20 000 1501 1498 1649 1500 1500 1500 150 000 150 000 20 000 20 000 150 000 150 000 150 000 150 000

38 31 42 55 81 69 31 44 44 165 44 50 109 88 96 74 155

121 b 123 b 111 b 128 b 165 b 179 b 101 b 147 b 137 b 198 b 81 b 113 b 175 b 146 b 139 b 122 b 190 b

9462 9311 9332 9289 9677 10 000 9677 10044 9915 3405 10 152 9742 9138 9527 9526 9677 7198

98 96.4 96.6 96.2 100.2 103.6 100.2 104.1 102.7 35.3 105.1 100.9 94.6 98.7 98.7 100.2 74.5

Core shear Core shear Core shear Indentation/buckling Indentation/buckling Indentation/buckling Indentation/buckling Core shear Core shear Core shear Core shear Core shear Indentation/buckling Core shear Core shear Core shear Core shear

a b c

Low/high load combination. Specimen not failed: tested for residual strength statically. High/low load combination.

83 92 69 73 84 110 70 103 93 33 37 63 66 58 43 48 35

S.D. Clark et al. / Composites Science and Technology 59 (1999) 471±486

477

4. Application of models 4.1. Single stress loadings 4.1.1. Initiation of fatigue damage The point of initiation of fatigue damage has been identi®ed from a cyclic creep model, reported elsewhere [14]. The approach adopted has been to subtract the cyclic creep response from the mixed creep/fatigue response to yield the initiation of pure fatigue. Typically most applied cyclic waveforms will comprise a small element of creep. In practice, it has been observed that the initiation of fatigue damage occurs where the rate of de¯ection versus number of cycles response just starts to increase near ®nal failure of the beam; see Fig. 5. The number of cycles to fatigue damage initiation calculated using this procedure for the test cases are shown in Table 4 and is shown graphically in Fig. 7. The main observation is that at higher stress levels, the number of cycles taken until the initiation of fatigue damage expressed as a percentage of the total fatigue life, is much smaller than for lower stress levels. In cases where the applied stress exceeds 66% of the ultimate static stress, initiation of fatigue damage occurs at under 50% of the number of cycles to failure, whereas for lower stress levels, the initiation of fatigue damage occurs at a much greater percentage of the total fatigue life, typically 70±90%. 4.1.2. Determination of fatigue modulus from experimental data To compare the theoretical sti€ness degradation model with experimental data, it is necessary to convert the overall de¯ection response into a change in sti€ness. Here it is assumed that the skins carry all the tensile/ compressive stresses and the core the shear stresses. Thus the fatigue response of a sandwich beam, where core shear is the observed failure mode, can be treated solely by the change in core shear modulus with number of cycles, assuming no interaction e€ects. Classical lin-

Fig. 7. S-N fatigue curve of damage initiation cycles and total cycles to failure.

ear beam theory incorporating a shear de¯ection is used to describe the static de¯ection response [15] and is outlined in Appendix A. The static response for sandwich beams has been found to approximate linear theory until close to failure [14]. The total de¯ection of a sandwich beam in ¯exure comprises a primary (bending) component and a secondary (shear induced) component. In core shear failure dominated fatigue, it may be assumed that the shear component alone degrades. The fatigue modulus can thus be estimated from externally measured de¯ections,  for a simply supported beam under a uniform load, w as follows total …n† ˆ bend ‡ shear …n†; wl2 where shear …n† ˆ 8AG…n†

…15†

4

5wl and bend ˆ 384D :

G(n) is the overall cyclic shear modulus, n the number of load cycles, l the beam span and D and A are the beam's ¯exural rigidity and the beam's e€ective cross sectional area, respectively, both de®ned in Appendix A. The cycle dependent shear de¯ection can therefore be calculated directly from experimental results after subtraction of the static bending de¯ection which is

Table 4 Experimental fatigue results and damage model constants Beam no. Applied stress ratio

C08 C09 C12 C16 C17 C18 C13 C11 C10 C06

0.565 0.547 0.674 0.604 0.621 0.626 0.477 0.714 0.680 0.453

Initial De¯ection at de¯ection initiation of fatigue damage

20.00 17.81 24.56 21.72 24.10 23.65 18.80 26.76 23.32 16.79

23.51 22.12 26.39 23.51 27.13 26.54 21.68 28.08 25.77 18.40

Cycles to fatigue damage initiation

18 301 52 315 1344 4601 4801 2201 122 341 1019 5322 25 896

Total Exponential fatigue model degradation constants cycles to failure

20 201 56 015 2344 5801 5801 3201 136 141 2419 6222 33 996

A

B

C

0.0834 0.1241 0.6475 0.5172 0.347 0.5343 0.3796 0.5033 0.2094 0.5492

599.5 402.9 77.2 96.7 144.1 93.6 131.7 99.3 238.8 91.0

0.0024 0.0011 0.0035 0.0031 0.0036 0.0034 0.0002 0.0024 0.0043 0.0005

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independent of the number of applied cycles assuming core shear degradation …16† shear …n† ˆ total …n† ÿ bend : Once the shear de¯ection is known the overall e€ective core shear modulus G(n) can be calculated: G…n† ˆ

wl2 : 8Ashear …n†

…17†

It has been shown from an earlier study [14] that the overall de¯ection response of a sandwich beam can be due to both fatigue and creep phenomena for many types of applied waveform (e.g. trapezoidal waveform incorporating cyclic elements (fatigue) and time at maximum constant load (creep)). Early on in the loading regime, up to a point described as the initiation of fatigue damage, creep processes are dominant whilst fatigue damage is only apparent after the initiation of fatigue damage. The initiation of fatigue damage, deduced from cyclic creep modelling, was found to occur where the rate of de¯ection starts to increase close to the end of the beams life [14]; see Fig. 5. For the normal range of fatigue frequencies (0.3±5.0 Hz), the component of creep de¯ection is small though not negligible compared to the de¯ection caused by fatigue processes. The overall e€ective shear modulus, G(n) therefore represents both a creep modulus and a fatigue modulus term: G…n† ˆ
Gc …n† ‡ Gf …n†;

…18†

where:
Gc …n† ˆ G0 ÿ Gc …n†: Gc(n) and Gf(n) are the creep and fatigue modulus respectively. Gf(n) already incorporates the static instantaneous modulus therefore the creep component only represents the di€erence between the static and transient creep modulus. Calculation of the creep modulus, published elsewhere [16] is similar to the approach used for the calculation of the fatigue modulus where the creep compliance is expressed as the inverse of the time dependent core shear modulus. In order to model the pure fatigue response, the creep component therefore has to be subtracted out from the overall response. The proposed approach to model the fatigue component of the de¯ection response, is to split up the fatigue response into two parts; the ®rst where the modulus remains unchanged from its initial static value until the initiation of fatigue damage, and the second where the fatigue modulus rapidly decreases until failure. The number of cycles to the initiation of fatigue damage is de®ned by nif whilst the total number of cycles to failure (in uppercase format implying failure has occurred) is, Nf. Thus up to the initiation of fatigue damage at nif cycles,

fatigue modulus, Gf(n) can be assumed to remain unchanged and equal the static modulus, G0. After nif cycles, Gf(n) rapidly reduces to its failure value at Nf cycles. The decrease in overall modulus, G(n) up to the initiation of fatigue damage can be deduced to be due to creep e€ects alone, i.e. if n 4 nf, G(n)=Gc(n) and can be calculated directly from Eq. (17) where shear is the shear de¯ection due to creep. Thus the fatigue modulus, Gf(n) can be calculated from an experimental de¯ection versus number of cycles plot using the following Gf …n† ˆ G0 ; Gf …n† ˆ

wl2 8Ashear …n†

ÿ Gc …n†;

where n4nif where n5nif

…19†

Eq. (19) represents the reduction of the fatigue modulus, initially at its static value to the initiation of fatigue damage, and then reducing after initiation till failure. The modulus degradation curves calculated directly from experimental data using Eq. (19) are shown in Fig. 8. It can be seen that the core shear fatigue modulus is assumed to be equal to the static modulus until the initiation of fatigue damage, whereas afterwards, it rapidly declines in a short number of cycles until failure. At higher stress levels, the rate of decrease of the modulus is greater earlier on in the beam's life. 4.1.3. Determination of fatigue model degradation parameters To determine the exponential parameters A and C of the sti€ness degradation model (Eq. (3)), experimental modulus degradation curves of the secondary region, for single step loading tests, are plotted as a modulus decrease versus number of cycles. Because fatigue damage only occurs after the initiation of fatigue damage, the cycle number, for the purpose of ®tting the parameters, is given by n ÿ nif. The exponential equation used to ®t the parameters is G…n ÿ nif † ˆ Ae…nÿnif †C :

…20†

An example of this approach is shown in Fig. 9. Suitable parameters calculated using least square ®t techniques

Fig. 8. Fatigue modulus degradation curves.

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479

Fig. 9. Exponential ®t to fatigue modulus degradation curve.

Fig. 11. Non-Linear S-N Curve based on fatigue modulus degradation.

were carried out at a variety of applied stress levels. The exponential coecient, A and the exponent, C were found to vary with the applied stress level in a non-linear manner; see Fig. 10. Here the maximum applied load is represented as a maximum induced core shear stress near the supports expressed as an applied shear stress ratio, r, i.e. applied stress/ultimate static shear stress. It is observed that there is a fair amount of scatter, especially for parameter A, and the form of the curves needed to ®t both parameters is not obvious. At r=0 it is expected that both parameters C and A will be zero, i.e. no degradation, whilst at r=1 it is not immediately evident what values the parameters should assume though the relation AeC=G0 holds. It was decided that as a ®rst approximation an exponential ®t to the parameters would show reasonable correlation satisfying the zero condition at stress ratio, r=0. This approach allows reasonable prediction of modulus degradation within the stress bounds used for testing. Further investigation needs to be carried out to identify the complete shape of the parameter curves at higher stress levels, a matter which will require further testing.

fatigue damage can be estimated using the non-linear power law S-N curve shown in Fig. 7. This estimated SN curve compared with the mean regression line used for general S-N characterisation in other published work [1±3] is shown in Fig. 11. It is clear that the non linear S-N model shows good correlation with the experimental fatigue data. It is observed that slight curvature is apparent in the non linear S-N curve. At low stresses, corresponding to a life of over 106 cycles, more curvature is observed suggesting the occurrence that a possible fatigue limit after 107 cycles is predicted. This trend is also observed with the experimental data from other published sources [1±3]. Therefore it is apparent that above 106 cycles, the chosen form of non-linear model is more suitable than the mean regression line which tends to give a conservative estimate of life.

4.1.4. S-N curve An S-N curve has been formulated based on the exponential shear modulus degradation approach using Eq. (5). The number of cycles up to the initiation of

4.1.5. Cumulative damage equations The three di€erent cumulative damage curves calculated from Eqs. (10), (12) and (14) for each of two stress levels are represented schematically in Fig. 12. One observation is that the ®rst model (Miner's rule), is linear as expected. The other two models are non-linear and display a ¯attening out of the damage curve at higher stress levels. What is also evident from the nonlinear models is that damage accumulates faster at higher loads than at lower loads. The exponential

Fig. 10. Variation of fatigue modulus degradation parameter `A' and `C' (Determined from experimental data).

Fig. 12. Cumulative damage models: r=0.7 and r=0.4.

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S.D. Clark et al. / Composites Science and Technology 59 (1999) 471±486

nature of the increasing damage just near the end of a beam's life is more consistent with observed behaviour (as shown in Figs. 5 and 8) than the linear case. 4.2. Multi-stress loadings 4.2.1. 2-step loading (A) Prediction of residual strength: The results from 2step loading tests (Table 3) show that all the beams did not fail at the end of the applied loading history and therefore had to be tested for their residual static strength in 4-point bending. The number of cycles at each load was calculated based on a previous batch of beams from single step loading tests (Table 4) and the combination of the two cycle duration was chosen to be equal or be greater than the estimated expected fatigue life. It can be concluded from the 2-step loading tests that because no fatigue failures occurred, the beams are slightly stronger than the previous batch. To compare the theoretical model based on shear modulus degradation with experimental results, the difference in batch strength has to be accounted for because the theoretical model parameters are based on single step loading tests (Table 4). This was achieved by changing the applied number of cycles to failure at each load for the single step loading tests to an assumed number of cycles to failure for 2-step loading results using a cycle shifting factor. The cycles to failure for the 2-step loading tests was assumed to be only slightly greater than the test cycles shown because, (a) some noticeable strength degradation was observed for some of the specimens and (b) two beams (not shown in Table 3) failed, hence failure for the main batch of beams tested was deemed to be immanent. For the calculations carried out, a shifting factor of 2.6 was used, i.e. increase in batch strength is equal to an increase in life by 2.6 times. The increase in life due to the increase in batch strength is proportional to an increase in core shear strength of approximately 2%, thus the di€erence between the two S-N curves correlating to each batch strength is small. The 2-step loading calculations for core shear modulus degradation, based on an adjusted number of cycles accounting for the variation in batch strength is shown in Table 5. The total number of cycles to failure at each load is estimated using the non linear S-N curve given by Eq. (5). The number of cycles up to the initiation of fatigue damage was estimated using a power law ®t to fatigue damage initiation results as shown in Fig. 7. The fatigue response up to the initiation of fatigue damage is assumed to be linear. If the number of cycles at the end of the ®rst load lies within the linear region, then the ratio of the number of cycles at the start of the second load, to the failure number of cycles at the second load, is proportional to the ratio of the ®rst applied cycles, to the failure number of cycles under the ®rst

load, i.e. n1/N1=n2/N2. If the ®rst number of cycles is after the initiation of fatigue damage, then the damage models need to be used to ®nd the e€ective number of cycles at the start of the second stress level. This is achieved by assuming that the amount of damage at the end of the ®rst load is equal to the amount of damage at the start of the second load. The number of cycles up to the initiation of fatigue damage is typically a large percentage of the total fatigue life, thus for all the experimental tests, the number of cycles due to the ®rst load (approximately 50% of Nf) was within the linear region. The residual modulus at the end of the application of the second load was calculated using the modulus degradation equation in Eq. (3). Residual strength was then estimated using a fatigue failure strain (equalling a static, linear limit core shear strain value, obtained from separate tests) of 3.8%. When comparing Tables 3 and 5 corresponding to experimental and theoretical predictions of two step loading, it can be found that there is generally good agreement. In particular, the pattern of residual strengths is the same; the order of the weakest beams displaying appreciable strength degradation is predicted well. The results are also encouraging in the sense that beams where no strength degradation is observed are also predicted to show negligible strength degradation. (B) Prediction of cumulative damage response: The accumulation of damage in beams according to the proposed cumulative damage models has also been investigated. Theoretical predictions for 2-step loading using the cumulative damage models corresponding to a low/high and a high/low loading combination are shown in Figs. 13 and 14, respectively. The number of cycles under the ®rst load has been chosen deliberately large (in each case 97.5% of the expected number of cycles to failure under the ®rst load) so as to purposely promote the initiation of fatigue damage under the ®rst load. The number of cycles under the second load is that required to induce failure, i.e. damage level=1. It can be observed that both nonlinear models predict that failure under a low/high loading combination occurs later than the linear rule (Model I) suggests, whilst the reverse is true for a high/low loading combination, i.e. high/low load combination more damaging. For all the models, the rate of damage accumulation per cycle is less at lower loads with a greater number of cycles to failure, than for higher loads with fewer cycles to failure. When considering the nonlinear models, Model III displays the smallest rate of change of damage in the ®rst phase of damage accumulation and the greatest rate of change of damage in the ®nal phase of damage accumulation. This is shown by the crossover of damage between Model II and Model III in Fig. 14. Model III gives the smallest fatigue life for a high/low loading combination and the greatest fatigue life for a low/high loading combination.

0.457 0.450 0.464 0.464 0.557 0.574

S01 b S02 b S08 b S09 b S04 b S05 b

a

0.452 0.455 0.564 0.566 0.476 0.463

0.575 0.572 0.663 0.695 0.684 0.688

S17 a S28 a S19 a S20 a S21 a S22 a

150 000 150 000 150 000 150 000 20 252 20 255

40 000 40 000 1500 1500 1578 1580

Number cycles at 1st load

20 000 20 000 1500 1500 1501 1498

150 000 150 000 20 000 20 000 150 000 150 000

Number cycles at 2nd load

56 818 56 818 56 818 56 818 7671 7672

15 152 15 152 568 568 598 598

Adjusted cycles at 1st load

7576 7576 568 568 569 567

56 818 56 818 7576 7576 56 818 56 818

Adjusted cycles at 2nd load

High/low loading combination. b Low/high loading combination.

0.575 0.579 0.693 0.689 0.678 0.685

2nd stress ratio

Beam 1st stress no. ratio

Table 5 Theoretical shear modulus degradation due to 2-step loading

259 454 319 089 217 326 214 479 21 610 14 956

14 641 15 443 2436 1332 1628 1534

Initiation cycles under 1st load

14 486 13 214 1396 1507 1824 1623

298 160 278 705 18 749 17 889 158 956 223 144

Initiation cycles under 2nd load

282 656 345 108 238 323 235 319 26 499 18 716

18 342 19 291 3319 1840 2242 2115

Cycles to failure at 1st load

18 159 16 648 1928 2079 2506 2236

323 226 302 851 23 172 22 166 176 493 244 458

Cycles to failure at 2nd load

3650 2741 460 502 725 916

266 996 237 863 3967 6844 47 051 69 169

Equivalent cycles at start of 2nd load

11226 10317 1028 1070 1294 1484

323814 294681 11543 14419 103869 125987

Number of cycles after 2nd load

50.00 50.00 49.95 49.97 49.97 49.79

18.45 46.44 50.00 50.00 50.00 50.00

Residual shear modulus (N/mm2)

99.99 99.99 99.91 99.93 99.95 99.58

36.90 92.87 100.00 99.99 100.00 100.00

%Residual shear strength

S.D. Clark et al. / Composites Science and Technology 59 (1999) 471±486 481

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S.D. Clark et al. / Composites Science and Technology 59 (1999) 471±486

Fig. 13. 2-step loading cumulative damage response: low/high load combination; r=0.4, 0.6.

Fig. 15. Comparison of 2-Step and 2-step block loading: low/high load combination; r=0.4, 0.6.

The e€ect of load sequence is also clearly demonstrated in Table 6, where the number of cycles under each load, for a low/high and a high/low load combination in Figs. 13 and 14, respectively, have been normalised so as to easily calculate the linear damage Miner's summation. Two approaches are shown in the ®rst two rows; (a) where the number of cycles at each load is normalised with respect to the total cycles to failure under that load; (b) where the number of cycles within the damage region, i.e. after nif, has been normalised with respect to the number of cycles expected to cause failure, i.e. Nfÿnif. The load is expressed as a induced core shear stress ratio, r, i.e.  app/ ult. Table 6 also shows in the last two rows, calculations of Miner's sum for each of the proposed cumulative damage models after the application of the two applied stress ratios, r=0.4 and r=0.6. Looking at the cycles normalised with respect to the damage region, Model I (based on Miner's rule), shows that the combined sum of the loadings is equal to one, independent of the load sequence. The nonlinear models clearly show that a low/ high load sequence gives a greater than unity life, with the reverse applying for a high/low loading sequence with a less than unity life. This is partly due to the shape of the damage curve which is dependent on the applied stress level, see Fig. 12. Additionally, the point of

fatigue damage initiation is non-linearly dependent with the applied stress level. It can be observed that at lower loads, the proportion of cycles in the fatigue damage region becomes relatively smaller. Thus at a high load, where damage initiates relatively earlier, an amount of damage after the ®rst loading level is equated to an equal amount of damage at the start of a lower second load where damage initiates relatively later. Thus even assuming linear damage, the nonlinear trend for the initiation of fatigue life e€ectively decreases the amount of residual normalised life. Miner's sum for Model III shows that the load sequence e€ect is only slightly stronger than for Model II, where failure occurs slightly earlier than Model II under a high/low loading combination.

Fig. 14. 2-step loading cumulative damage response: high/low load combination; r=0.6, 0.4.

Fig. 16. Comparison of 2-Step and 2-step block loading: high/low load combination; r=0.6, 0.4.

4.2.2. 2-Step block loading As a further step towards understanding the e€ects of load sequence, an analysis of the e€ect of multi-step block loading, i.e. repetitions of load sequences was undertaken using the cumulative damage models. In this case, a 2-step load combination was compared with a 2step block loading combination. The 2-step block load combination used was a single repetition of the chosen 2-step loading combination, e.g. low/high/low/high. It was assumed that the after the point of fatigue damage

S.D. Clark et al. / Composites Science and Technology 59 (1999) 471±486

initiation, the beam was loaded at 50% and 25% of the beams expected life at that load for 2-step and 2-step block loading combinations respectively. The results for a low/high and a high/low loading combination are shown in Figs. 15 and 16, respectively. The results show that the linear damage rule (Model I), as expected in both 2-step and 2-step block loading conditions, yields the same life with a Miner's summation of unity at failure. Both nonlinear models (Model II and Model III) follow the same general load sequence trends described earlier. Interestingly, Model II appears to give the same Miner's summation at failure for both multi-step and multi-step block loadings. For Model III however, repetitions of low/high or high/low loads tend to have a slight averaging e€ect on fatigue lifetime, thus a high/ low 2-step load combination will have a slightly reduced life when compared to a high/low 2-step block loading condition. This can be easily deduced by comparing the Miner's summations for the two forms of loading; in this case 0.787 compared to 0.808 for 2-step and 2-step block loading conditions respectively. Conversely, a low/high 2-step load combination will have a slightly increased life compared to a low/high 2-step block loading condition. 5. Discussion 5.1. Fatigue scatter There is an observable amount of scatter in the S-N data which is typical of fatigue data in general. Scatter is most likely to be a result of the variation in mechanical material properties in the foam cores and/or manufacturing inconsistency. The measured maximum percentage error in foam densities about the quoted manufactures nominal density (calculated by weighing the foam) was approximately ‹2% for the C70.130 foam. Due to the scatter in the experimental data, there can be diculty in obtaining experimental load sequence trends. Generally speaking for two step loading, the change in load step has to applied after the initiation of fatigue, for non-linear behaviour to be observed (i.e. D > 0). Fatigue damage typically occurs in only a short

483

number of cycles relative to the time taken to cause initiation of fatigue damage. Thus when testing a sandwich beam whose initial strength is not precisely known (due to variabilities in materials and manufacturing), it is extremely dicult to predict this secondary region of rapidly accumulating damage where very small changes in initial strength can cause relatively large di€erences in fatigue life; a trend highlighted by the shallow gradient of the S-N curve. 5.2. Sti€ness degradation model Fatigue behaviour for core shear failures has been modelled using a sti€ness degradation approach assuming that the core shear modulus degrades with increasing number of cycles. Up to the initiation of fatigue, the core shear fatigue modulus is assumed to be equal to the instantaneous static value (i.e. manufacturers quoted value of shear modulus). Separate residual strength tests of beams fatigued at a single load level have demonstrated that there is virtually no decrease in strength in this region and the observed small reduction in core shear modulus apparent in this region has been found to be due to cyclic creep mechanisms. The chosen approach of splitting the response into two parts (an initiation period and a propagation period) therefore appears to be justi®ed in this case. Whilst there is a small amount of permanent core shear modulus reduction due to ``secondary'' creep behaviour (irrecoverable creep) before initiation of fatigue damage, it is assumed to be negligible compared to the magnitude of modulus degradation due to fatigue damage after initiation. This supports the conclusion that for the frequency ranges considered, fatigue damage mechanisms are almost totally responsible for ®nal failure of the beam (i.e. in terms of beam lifetime, fatigue damage mechanisms are totally attributable to ®nal failure of the specimen). After the initiation of fatigue damage, an exponential law was found to ®t the experimental core shear modulus degradation trends reasonably well. The S-N equation based on the modulus degradation law and the strain failure criterion (fatigue failure strain is equal to static failure strain), correlates very well with the experimental S-N curve, thus supporting the chosen

Table 6 Cumulative damage predictions for 2-step loading Low/high load combination

High/low load combination

r=0.4 r=0.6

r=0.6 r=0.4

Model I Model II Model III % total cycles (% n/Nf) % cycles within damage region (% (n ÿ nif)/(Nfÿnif)) Combined Miner's sum based on total cycles Combined Miner's sum based on cycles within damage region

97.5 35.8

14.4 64.2 1.119 1.000

21.2 94.4 1.187 1.302

23.2 103.2 1.207 1.390

Model I Model II Model III 97.5 89.0

0.42 11.0 0.979 1.000

0.29 7.5 0.978 0.965

0.21 5.5 0.977 0.945

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S.D. Clark et al. / Composites Science and Technology 59 (1999) 471±486

form of degradation equation and corresponding underlying assumptions. 5.3. Damage models Non-linear fatigue damage is assumed to emerge after the initiation of fatigue damage where the residual life is based purely on equating equivalent damage levels on the corresponding cumulative damage curves at changes in the load step. Thus fatigue lifetime will only be signi®cantly in¯uenced by the magnitude and sequence of the load levels experienced after the initiation of fatigue damage. For multi-step loading, the fatigue life is mostly dependent on the load pattern at the initiation of fatigue damage. Two non-linear cumulative damage models have been compared with a linear damage rule, all of them applied after the initiation of fatigue damage. The non-linear models predict that a high/low load combination results in a reduced life compared to the linear rule. The reverse applies for a low/high load sequence. The two parameter cumulative damage model, (Model III), displays the greatest nonlinearities. The main observation was that at higher stress levels a ¯attening out of the damage curves for the two non-linear models is observed. This suggests that damage may accumulate more uniformly at higher stresses. When applied to multi-step block loading, Model III predicts that repeated sequences of loadings tend to result in less extreme behaviour than a single load sequence combination. Whilst the load dependency appears to be strong after the initiation of fatigue damage, it only represents a small percentage of the overall specimen's life, thus the e€ect is that the load sequence trend over the whole specimens life becomes less perceptible. Experimental results of 2-step loading have indicated the possible existence of non-linear fatigue damage behaviour, where the order of load sequence a€ects the resultant fatigue life. This trend has been found to be due to non-linear damage behaviour after the point of fatigue initiation. One way in which this trend manifests itself is through decrease in strength after the appropriate number of fatigue load cycles. It has been shown that the cases where residual strength decreased signi®cantly from static values were those where a high/ low load combination was used. In other cases, especially in all low/high load cases, there was no marked strength decrease. This is consistent with the trend given by the nonlinear cumulative damage models where a high/low load combination is shown to be more damaging than a low/high load combination.

the core material. Experimental tests under both single step and two-step loading have been undertaken where core shear was the failure mode. The results show that a point of fatigue damage initiation can be de®ned which is typically close to ®nal failure of the specimen. The proposed sti€ness degradation model demonstrates that, in accordance with experimental results, the fatigue core shear modulus rapidly decreases at an increasing rate in an inverse exponential manner close to failure. Preliminary experimental ®ndings for 2-step loading, especially those related to residual strength, indicate that there may be a slight load sequence e€ect. Theoretical cumulative damage models derived from the sti€ness degradation model have also been proposed which show the same load sequence e€ect where a high/ low load combination is more damaging than a low/ high load combination. Acknowledgements The authors wish to thank the EPSRC for providing funding for research under the `High Speed Craft' managed programme being overseen by the MTD and Marinetech South. Appendix A. Static sandwich beam equations Classical linear beam theory incorporating a shear response is often used to predict sandwich beam de¯ection behaviour [15]. In this case, the de¯ection response is assumed to be linear though in practice the experimental response is only approximately linear until a point very close to failure; see Fig. 17. Thus up to failure, correlation between theory and experimentation is extremely good and even at failure, failure loads will be predicted well though will de¯ections will be slightly underestimated. The main assumptions of the theory are that: (a) (a) the faces are thin so that the second moment of area about their own neutral axis is negligible.

6. Conclusions A unique fatigue model for sandwich structures has been proposed based on the degradation of sti€ness in

Fig. 17. Typical static core shear failure load-de¯ection plot.

S.D. Clark et al. / Composites Science and Technology 59 (1999) 471±486

(b) (b) the core makes a negligible contribution to the ¯exural sti€ness of the sandwich hence the shear stress is approximately uniform throughout the core. The beam length is l with beam width b and core depth c. Other dimensions are de®ned in Fig. 18. Skin Properties: Skins moduli and stresses (tension or compression), E1, E2, and direct stresses,  1,  2, respectively. Top and bottom skin thickness' are t1 and t2, respectively (subscripts 1, 2 refer to top and bottom skin, respectively). The distance between the neutral axis of the two facings is d and t is the average skin thickness. Core Properties: Ultimate core shear stress,  c and core shear modulus, G. Core thickness, c. Tensile and compressive core moduli, Et, Ec and direct tension and compression core stresses  t,  c, respectively. Sandwich Beam Flexural Rigidity, i.e. beam sti€ness: For two faces of unequal thickness   E2 t2 =E1 t1 : …A:1† D ˆ bd2 E1 t1 1 ‡ E2 t2 =E1 t1 Skin Direct Tension or Compression Failure: use ordinary beam theory. 1;2

Mz1;2 E1;2 : ˆ D

…A:2†

81;2 D : pˆ 2 bl z1;2 E1;2

Fig. 18. Typical sandwich beam cross section.

The distance from the neutral axis to the top face, z1 can be expressed as z1 ˆ

E2 t2 t1 d‡ : …E1 t1 ‡ E2 t2 † 2

…A:4†

The distance from the neutral axis to the bottom face, z2 can be expressed as z2 ˆ

E1 t1 t2 d‡ : …E1 t1 ‡ E2 t2 † 2

…A:5†

Core Shear Failure: Assume shear stress constant through depth of core c ˆ

Q : bd

…A:6†

For a simply supported beam under a uniform load; shear force, Q=wl/2, hence w can be calculated in terms of core's ultimate static shear stress,  c. In terms of the applied pressure at failure, p in (Mpa) pˆ

2dc : l

…A:7†

Sandwich beam de¯ection equation: The total de¯ection is the summation of bending and shear components total ˆ bend ‡ shear :

For a simply supported beam under a uniform load; bending moment, M=wl2/8 and z is distance from neutral axis of the beam to the edge of the skin under consideration. Hence the applied uniform load, w can be calculated in terms of skin's ultimate static direct stress,  f and converted to a uniform failure pressure, p=w/b (in MPa).

485

…A:8†

For a simply supported beam under uniform load, w, the de¯ection at midspan is ˆ

5wl4 wl2 ‡ ; 384D 8AG

…A:9†

where A=bd2/c and G is the core shear modulus.

…A:3† References [1] Bergan PG, Buene L, Echtermeyer AT, Hayman B. Assessment of FRP sandwich structures for marine applications. Marine Structures 1994;7:457±73. [2] Bertelsen WD, Eyre M, Sikarskie DL. Veri®cation of the hydromat test system for sandwich panels. Sandwich Constructions 3, vol. 1. UK: Engineering Materials Advisory Service, 1995. p. 269±80. [3] Caprino G, Teti R, Messa M. Long-term behaviour of PVC foam cores for structural sandwich construction. Sandwich Constructions 3, vol. 2. UK: Engineering Materials Advisory Service, 1995. p. 813±24. [4] Chevalier JL, Creep fatigue and ®re resistance of chemical tank sandwich cores. Journal of Reinforced Plastics and Composites 1994;13:250±61. [5] Olsson KA, LoÈnnoÈ A.Sandwich constructions±recent research and development: GRP sandwich technology for high speed marine vessels. Sandwich Constructions 2. In: Weissman-Berman

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[11] Hwang W, Han KS. Fatigue of composites fatigue modulus concept and life prediction. Journal of Composite Materials 1986;20:155±65. [12] Miner M. Cumulative damage in fatigue. Journal of Applied Mechanics Part A 1945:159±64. [13] Allen HG, Shenoi RA. Flexural fatigue tests on sandwich structures. Sandwich Constructions 2. In: Weissman-Berman D, Olsson KA, editors. Proceedings of the 2nd International Conference on Sandwich Construction, vol II. UK: Engineering Materials Advisory Services, 1992. p. 499±517. [14] Clark SD. Long term behaviour of FRP structural foam cored sandwich beams. Ph.D Thesis, University of Southampton, UK, 1997. [15] Allen HG. Analysis and design of structural sandwich panels. London: Pergamon Press, 1969. [16] Shenoi RA, Allen HG, Clark SD. Cyclic creep and creep fatigue interaction in sandwich beams. Journal of Strain Analysis 1997;32(1):1±18.