Fracture Mechanics Analysis of Reinforced DCB Sandwich Debond Specimen Loaded by Moments

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1 Downloaded from orbit.dtu.dk on: Oct 07, 08 Fracture Mechanic Analyi of Reinforced DCB Sandwich Debond Specimen Loaded by Moment Saeendran, Vihnu; Berggreen, Chritian; Carlon, Leif A. Publihed in: A I A A Journal Link to article, DOI: 0.54/.J Publication date: 08 Document Verion Peer reviewed verion Link back to DTU Orbit Citation (APA): Saeendran, V., Berggreen, C., & Carlon, L. A. (08). Fracture Mechanic Analyi of Reinforced DCB Sandwich Debond Specimen Loaded by Moment. A I A A Journal, 56(), DOI: 0.54/.J General right Copyright and moral right for the publication made acceible in the public portal are retained by the author and/or other copyright owner and it i a condition of acceing publication that uer recognie and abide by the legal requirement aociated with thee right. Uer may download and print one copy of any publication from the public portal for the purpoe of private tudy or reearch. You may not further ditribute the material or ue it for any profit-making activity or commercial gain You may freely ditribute the URL identifying the publication in the public portal If you believe that thi document breache copyright pleae contact u providing detail, and we will remove acce to the work immediately and invetigate your claim.

2 Energy Releae Rate and Mode-Miity of Face/Core Debond in Reinforced Sandwich Specimen Loaded by Moment Vihnu Saeendran and Chritian Berggreen Technical Univerity of Denmark, Kg Lyngby, DK web page: Leif A. Carlon Florida Atlantic Univerity, Boca Raton Florida 3343, USA web page: Abtract Analytical epreion for energy releae rate and mode miity phae angle are derived for a andwich compoite double cantilever beam fracture pecimen with the face heet reinforced by tiff plate. The andwich beam i conidered ymmetric with identical top and bottom face heet. Only pure moment loading i conidered. J-integral coupled with laminate theory i employed to derive cloed form epreion for the energy releae rate in term of applied moment, geometry and material propertie. A calar quantity ω i obtained to epre mode miity phae angle. It i hown that ω i independent of applied loading condition. The value of ω i found to be moderately influenced by reinforcement thicknee. Nomenclature ψ mode-miity phae angle G energy releae rate J J- integral a pre-crack length PhD Fellow, Department of Mechanical Engineering Aociate Profeor, Department of Mechanical Engineering Profeor, Department of Ocean and Mechanical Engineering

3 Γ integration path for J integral 0 ε - laminate mid-plane train κ - laminate mid-plane curvature σ ij tre tenor M d moment applied on debonded beam M moment applied on ubtrate beam E f Young modulu of face heet E c Young modulu of core E r Young modulu of reinforcement layer h f thickne of face heet h c thickne of core h r thickne of reinforcement layer K tre intenity factor β - Dundur bimaterial parameter ε ocillatory inde e d neutral ai ditance (debonded beam) e neutral ai ditance (ubtrate beam)

4 I. Introduction Face/core interface debonding i a eriou failure mode that affect the performance of a andwich tructure. Debond (face and core eparation) can occur during liquid rein proceing due to inadequate wetting of the face/core interface region which reduce the adheive trength between face and core. Face/core debond may alo occur due to ervice load uch a wave lamming, impact and fatigue cycling. Debond may propagate along the interface or kink into the core. The propenity of the crack to propagate i determined by the local tre tate at the crack tip for a given loading condition. The tre intenity factor at the crack tip for a given loading condition can be epreed in term of a mode-miity phae angle (ψ) which quantifie the ratio of hear to normal loading at the crack tip. Determination of the interface fracture reitance i vital from a deign perpective. There are variou eperimental method developed to determine the interface fracture toughne uch a the Cracked Sandwich Beam (CSB) [], Double Cantilever Beam (DCB) [], Tilted Sandwich Debond (TSD) pecimen [3], the Three-Point Sandwich Beam (TPSB) [4], Mied-mode Bending (MMB) pecimen [5] and Single Cantilever Beam (SCB) andwich pecimen [6]. Mot of the devied eperimental tet method were inpired by fracture tet method developed for laminate compoite. For intance, the MMB tet method developed for delamination teting [7], [8] wa etended to andwich compoite [5] [9]. The SCB andwich pecimen i a imple tet et-up for determining mode I fracture toughne of face/core interface [0]. However, appropriate izing of the pecimen mut be undertaken to enure that the face/core crack propagate along the interface [] at mode I loading. Effort are underway to implement the SCB andwich pecimen a a tandard tet method for mode I fracture toughne characterization []. Due to the high elatic mimatch acro the interface in andwich compoite, the face/core crack i inherently mied mode. A full characterization of the face/core interface inevitably require teting over a wide array of mode-miity phae angle. Therefore, it i deirable to control the mode-miity during the tet. A relatively recently developed tet method for delamination teting i the Double Cantilever Beam loaded with Uneven Bending Moment (DCB-UBM) developed by Sørenen et al. [3]. Thi method wa recently etended to andwich compoite by Øtergaard et al. [4] and Lundgaard- 3

5 Laren et al. [5], and i chematically illutrated in Fig. In thi method, it i poible to perform a fracture tet at a deired mode-miity by controlling the moment M and M applied to the pecimen end. Figure. DCB-UBM pecimen loaded with edge moment Reference i made here to the crack element approach by Suo and Hutchinon [6] who developed a fracture mechanic analyi approach for a bi-layer element and Kardomatea et al. [7] who etended thi procedure to a cracked andwich element. Thee author conidered only in-plane (aial) force and moment couple acting on the edge of the pecimen. An analytical epreion for the energy releae rate, G, wa obtained through the J-integral. The mode decompoition wa performed uing the tre intenity factor K I and K II, derived analytically ecept for a ingle calar parameter ω, which wa etracted from the numerical olution of one loading combination. Sandwich panel with thin faceheet (in the range of 0.5 mm) are not uncommon, epecially in the aircraft indutry. Fracture characterization of uch andwich compoite poe many problem uch a load application to the debonded face heet which, if thin, will undergo large nonlinear deflection and rotation. A method to reduce diplacement i to reinforce one or both face with tiff layer named doubler. Thi method wa adopted by Lundgaard-Laren et al. [5], who bonded tiff teel plate to both faceheet to reduce the rotation. In thi paper, epreion for the energy releae rate and mode-miity phae angle are derived for a reinforced DCB-UBM fracture pecimen loaded by pure edge moment. The mode-miity phae angle (ψ) quantifie the ratio between mode II and mode I tre intenity factor. II. Analyi of Sandwich Fracture Specimen The andwich pecimen conidered here (ee Fig. ) conit of five layer, two compoite face heet laminate labeled and, the core, and two reinforcing plate of thickne h r, bonded to each of the face heet. Typically, the face heet are 4

6 compoed of multi-directional laminate with plie arranged in a ymmetric and balanced way. Analyi of uch a andwich element i implified by homogenizing the laminate into a pecially orthotropic compoite layer of the ame thickne a the laminate, and tiffne E, E, ν, ν and G. The approach preented here, however, aume all layer are iotropic with Young modulu E and Poion ratio ν. Tranformation of orthotropic elatic contant of the laminate face heet into iotropic contant i dicued in Appendi B. Figure. Sandwich beam element with reinforcing doubler layer of thickne hr Figure 3 how the uperpoition cheme ued for the analyi of DCB-UBM pecimen. The original configuration i hown in Fig. 3a. By adding the un-cracked configuration ubject to pure moment per unit width, M 3, a hown in Fig. 3b, the force and moment configuration hown in Fig. 3c i obtained. A indicated in Fig. 3c, beam #, referred to a the debonded beam, conit of the top face heet and reinforcement layer (thickne, H = h r + h f) and beam # referred a the ubtrate part conit of the layer beneath the pre-crack i.e. core, bottom face heet and bottom reinforcement layer (thickne, H = h c + h f + h r). The intact portion right of the crack front compriing of both face heet, reinforcement layer and the core i referred to a the bae part (thickne, H 3 = h r + h f + h c). Hence, the two ytem will have ame energy-releae rate and tre intenity factor. Thi analyi follow the principal approach performed by Suo and Hutchinon for a bi-material interface [6]. The DCB-UBM pecimen i loaded by pure moment per unit width, M and M, applied to the left edge a hown in Fig.. Hence, there are no aial in-plane force or tranvere hear force acting on the pecimen. In ome cae, aial load are acting on the ection. Suo and Hutchinon [8] conidered aial load P, P and P 3 per unit width. P and P act on the 5

7 left edge and P 3 act on the right edge. The influence of thee force on the crack loading i included in their analyi, and it i poible to conider uch load alo in the preent analyi. For the pecific tet pecimen conidered here (DCB-UBM), however, there i no aial force preent, and hence i not conidered in our analyi. (a) = (b) (c) Figure 3. Superpoition cheme of andwich geometry The moment acting on the debonded arm, M d, and aial force, P can be epreed in term of M 3 a: P = c M 3 M = M cm d 3 3 (a) (b) () where epreion for c and c 3 are obtained from tre analyi of each beam provided later. Notice that the three original loading parameter are reduced to two independent variable, P and M d. From equilibrium, M * = Md + P () 6

8 where = ' H3 δ δ (3) (3a) and δ ' = H δ (3b) The ditribution of tre in each ub-beam can be determined from laminate beam theory [9] where each part of the andwich beam i conidered a multi-layered beam (ee Fig. 3). The thickne of the debonded beam # i H = h f + h r and of the ubtrate beam # i H = h c + h f + h r. The force and moment (per unit width) are given by [9]: N A 0 = ε+ Bκ (4a) (4) M B 0 (4b) = ε+ Dκ (3b) where N and M are the force and moment reultant, and coupling and bending tiffnee (A, B and D) are defined a: 0 ε and κ are the mid-plane train and curvature. The etenion, n A = Ek(yk y k ) k= n B= (y y E k ) k k k= n D = (y 3 y 3 ) 3 Ek k k k= (5a) (5b) (5) (5c) where the y-ai i referenced to the geometric mid-plane (y = 0). k i the layer inde k =,...n, where n i the number of layer. y k i the y-coordinate of the interface between layer k and k+. Note that y 0 = -h/ where h i the total laminate thickne. E i the elatic modulu in the - direction for ply k. For plane train, k k k / ( k) E = E ν while for plane tre, E k = E. An eample of the layer coordinate, (y k) for the intact part of the pecimen (#3 in Fig. 3c) i hown in Fig. 4. k 7

9 The tre in each layer i: Figure 4. Layer definition for the intact part (#3) of the andwich pecimen ( σ ) E k = kε (6) where the train i given by: ε = ε 0 + κ y (7) Conider firt the configuration hown in Fig. 3b. Figure 4 how the layer coordinate. For the pure bending cae, ubtituting (N = 0) in Eq. (4), provide the mid-plane train, and the y-coordinate of the neutral ai: 0 Bκ ε = (8a) (8) A y NA = B A (8b) For implicity, the andwich beam i aumed to be ymmetric (B = 0). For thi cae, Eq. (4b) yield, =. Rearranging and ubtituting for ε = 0 in Eq. (7) give, 0 M Dκ ε M3 = κ y = D y (9) 8

10 Subtituting ε in Eq. (6) to obtain tre: ( ) 3 obtained by integrating the tre ( σ ) k M σ = E k k y. The force, P, acting on debonded part (beam #), Fig. 3, i D over the cro ection a: H3/ P= σ dy= cm3 (0) h c / Integration yield, ( 4 3) + r( 5 4) Ef y y E y y c= () D where the ply coordinate are illutrated in Fig. 4. The moment M d, Fig. 3, acting on beam # (debonded) i given by: H3 h c Md= M σ y + δ dy h c (a) () Md = M cm 3 3 (b) where h c E f h 3 h 3 + δ c c hc h c = h h c 3 D + f + f hc 3 E H δ r 3 h + c H 3 h h c + f h D + + f (3) III. J-integral calculation The current analyi i carried out in the ambit of Linear Elatic Fracture Mechanic (LEFM) regime. In order to obtain the energy releae rate for a pre-cracked andwich element reinforced with tiff doubler layer, the J-integral approach i choen. 9

11 Figure 5. J-integral path in reinforced andwich beam J-integral wa calculated for the cloed path hown in Fig. 5 uing the general epreion [0] : J = Wdy n. ui Γ σij j ds (4) where σ ij i the tre tenor, ds i a length increment along the cloed path Γ, n j i an outward normal vector to the cloed contour and u i i the diplacement vector. W i the train energy denity, W = σε. The J-integral i non-zero only along the vertical path near the left edge marked Γ Γ 3 and Γ 9 - Γ 0. For the horizontal path dy= 0 and, the normal vector i directed along the y-ai: σ ijn j = 0, making no contribution to J. Furthermore, the vertical path (Γ 4 Γ 8) along right edge do not contribute to J a no load act on that edge (ee Fig. 5). The J-integral i evaluated for all layer and ummed up: J 0 = E pσ dy p= Γ (5) According to Eq. (5) J-integral i calculated from the tre σ which i due to the acting moment and force (ee Fig. 5). The total energy releae rate then become: J= G= JDebonded + JSubtrate = J+ J+ J3+ J9+ J0 (6) J through J 3 and J 9 - J 0 are calculated from the tre in each layer. A detailed derivation of the J integral i provided in Appendi A, which yield, 0

12 L V V V3 L V G = P M ( ) ( ) ( Eh) d + H H H H + Eh Eh d d V L 3 V M P 3 d ( Eh) Hd ( Eh) H H d (7) Thi equation i derived in Appendi A. The above epreion for G i re-arranged to obtain a quadratic form in P and M d imilar to [7] a follow: J = G = a P a M a PM + d 3 d (8) where L V V V a = ( Eh) ( Eh) H ( Eh) H (9a) (9) d L V a = + (9b) H H d V L3 V a = 3 3 (9c) H ( Eh) Hd ( Eh) H d IV. Mode-Miity epreion The energy releae rate may be epreed in term of a comple tre intenity factor (K = K + ik ) [3, 4] a: G= B K (0) where i = and where ε, the ocillatory inde i epreed a: ( κ + ) + G ( κ + ) Gc f f c B= () 6G G coh f c πε

13 β ε = ln π + β () The Dundur parameter β, i given by ( κ ) G ( κ ) ( κ + ) + G ( κ + ) Gf c c f β = Gc c c f (3) G f and G c are the hear moduli of the face and core. κ ( 3 4ν ) ( ν ) = + for plane train and κ = 3 4ν for m m m plane tre condition. ν m i the Poion ratio, m = and for upper face heet and core repectively. Subtituting energy releae rate, G from Eq. (8) in (0) yield: ( a P a M d a PMd ) K = 3 B + (4) m m There are two poible root for K in Eq. (4). The root for K include both real and imaginary part. Kardomatea et al. [7] found the root of a imilar equation following the approach of Thoule et al. [3] and Hutchinon et al. [6]. Therefore, eploiting imilar argument the comple tre intenity factor K can be written a: ( d ) K= ap a i + bm a h ε (5) B f It hould be noted that Eq. (4) i of ame form a in [7]. For the firt root, the comple number a and b are defined [8]: a= e i ε b= ie i ( ω+ γ ) (6) where a inγ = 3 (7) aa It i required that a and b are independent of loading for the derivation of cloed form olution of mode-miity. Thu, by electing the firt root, the parameter ω in Eq. (6) become dependent only on geometry and material propertie of the reinforced andwich pecimen but not on loading. Subtituting a and b in Eq. (5) lead to:

14 ( d ) K= K+ ik= P a ie i γ M a h f i ε e i ω (8) B The definition of the mode-miity phae angle follow Hutchinon and Suo [8] for a bimaterial interface crack. The mode miity phae angle ψ i defined a: Im[K h iε ] tan f ψ = Re[K h iε f ] (9) where the real and imaginary part of the argument are [3]: Re K h iε f = P acoω Md ain( ω γ) B + + (30a) (30) Im K h iε f = P ainω Md aco( ω γ) B + (30b) Note that the near tip ocillation i uppreed by thi definition of ψ, and that Re K h i f ε = K coψ and Im K h i K in f ε = ψ. An epreion for the phae angle ψ can be obtained from Eq.(9) and (30): ( ) ( ) in co tan ψ λ ω ω+ γ = λ co ω + in ω + γ (3) where P M λ = (3) d a a The parameter λ incorporate the influence of tiffened face heet through a and a. The parameter ω can be epreed in term of the phae angle a: ( ) coγ+ λ+ inγ tanψ ω= tan λ+ inγ coγ tanψ (33) 3

15 V. Calculation of ψ and ω Finite element analyi (FEA) of the DCB-UBM pecimen combined with a method to etract the tre intenity factor called the Crack Surface Diplacement Etrapolation (CSDE) method [4] i employed here to calculate ω and ψ. Twodimenional plane train model of andwich pecimen were made in ANSYS [5], compriing io-parametric 4-node (PLANE 4) and 8-node (PLANE 8) element. A highly dicretized meh wa ued near crack tip (ee Fig. 6). The face heet, core and reinforcement layer are conidered linear elatic and iotropic. The PLANE 4 element were ued at the crack tip with a minimum element ize of mm. Thee element were ued to capture the large train gradient encountered at the crack region. The mode-miity phae angle (ψ) i etracted from the near-tip crack flank diplacement in the following form: δ ψ= tan εln + tan ε δ y h ( ) (34) where i ditance behind the crack tip, ε i the ocillatory inde (Eq. ) and h i a characteritic length, which i taken a the face heet thickne, h = h f. The CSDE method i implemented a a ubroutine in the commercial FE-package ANSYS and employ crack flank opening and liding diplacement (δ y and δ ) over a region very cloe to behind the crack tip. The energy releae rate i given by [4]: πg ( 4 m + ε ) G ( CSDE = δ + δ y ) k ( m + ) (35) = for plane train and k m = (3 4 ν m )/( + ν m ) for plane tre, ν m i Poion ratio. m = and for face and where k m (3 4 ν m ) core. G and G are the hear moduli of the face and core material repectively. FEA i performed on both un-reinforced and reinforced DCB-UBM andwich pecimen. FEA reult for un-reinforced pecimen are compared to analytical epreion derived here for the energy releae rate G anal (Eq. (7)) and epreion available in literature for un-reinforced pecimen [7]. The material propertie of face and core employed in the analyi are provided in Table. In the econd part of analyi, a reinforced andwich DCB-UBM pecimen with a oft core (PVC H45 foam) i conidered. 4

16 Figure 6. FE-model of reinforced DCB-UBM andwich pecimen Table. Material propertie of face heet and core [9] Core H45 H00 H50 Aluminum foam core Young modulu, E c [MPa] Shear modulu, G c [MPa] ,630 Poion ratio, ν c Face heet Aluminum face E-gla fibre - DBLT-850(0/45/90/-45) Young modulu, E f [GPa] Shear modulu, G f [GPa] Poion ratio, ν f

17 V. (A). Unreinforced DCB-UBM Sandwich Specimen Two unreinforced andwich configuration compriing of mm thick aluminum face heet, and a 0 mm thick oft core (PVC H00) and a tiff core (aluminum foam) were choen to benchmark the analytical epreion. For both cae, refer to Table for material propertie. A crack length, a = 00 mm wa ued with a ufficiently long pecimen to reduce the edge effect (c = 300 mm). The mode-miity of a DCB-UBM pecimen i changed by altering the ratio of the moment M and M applied to the edge (moment ratio MR = M /M ) (Fig. ). The MR value, thicknee and material propertie were taken from [7] in-order to compare the energy releae rate and mode-miity reult to the reult obtained herein (Eq. 7 and 9). Such a direct comparion i made by making the reinforcement layer modulu equal to that of the face heet and making the um of each face heet thickne and reinforcement thickne equal to the face thickne analyzed in [7]. The reult are eamined over a range of moment ratio (MR). Table and 3 lit energy releae rate reult for a large range of moment ratio (MR). Cloe agreement between numerical (G CSDE) and analytical (G anal,) reult i noted. The current reult for G alo agree with [5] and [7]. Note that reult from [7] are compared here with moment loading only. It i furthermore noted that the parameter ω remain relatively contant for each cae. For the PVC core andwich, the larget deviation of ω from the average value i 0.5% wherea the deviation for the tiffer aluminum foam core i below.%. It hould be further pointed out that the phae angle reult preented in Table and 3 compare well with thoe publihed earlier in [7]. Table. G, ψ and ω reult for unreinforced DCB-UBM andwich pecimen with PVC H00 core Moment Ratio, MR M [Nmm] M [Nmm] G anal [N/mm] G CSDE [N/mm] G [N/mm] [7] ψ[deg] ω[deg]

18 Table 3. G, ψ and ω for unreinforced DCB-UBM andwich pecimen with Aluminum foam core Moment Ratio, MR M [Nmm] M [Nmm] G anal [N/mm] G CSDE [N/mm] G [N/mm] [7] ψ[deg] ω[deg] V. (B). Reinforced DCB-UBM Sandwich Specimen During fracture characterization tet of unreinforced andwich pecimen, eceive deformation of either crack flank will violate Linear Elatic Fracture Mechanic (LEFM). Reinforcing the fracture pecimen with tiff doubler layer prevent eceive crack flank rotation (ee Fig. 7). Moreover uch layer will make it eaier to attach loading tab to pecimen for eperimental teting. The parameter ω i computed for a reinforced DCB-UBM andwich pecimen with a oft PVC foam core (H45). A earlier, aluminum face heet were choen (h f = mm). Steel reinforcement layer were choen (E = 0 GPa, ν r = 0.3) with a thickne h r = 6 mm [5]. The total length of the pecimen wa L = 500 mm with a crack length, a = 00 mm. Reult for the reinforced pecimen are preented in Table 4. For the range of moment ratio eamined, the phae angle (ψ) varied from 8.74 to 77.3, while the calar parameter ω, remained nearly contant with an average of (±.5 %), ee Table 4. An advantage of a ω parameter that i independent of loading, i that the mode-miity phae angle may be computed uing a ingle ω value. To further eamine the parameter ω, the phae angle (ψ) wa calculated for a range of MR uing Eq. (3) with the average value of ω = (Table 4). The reult for ψ in Table 4 how that ψ value obtained uing a fied ω value (ψ*) cloely match the one from FEA. Figure 7. Sandwich DCB-UBM pecimen reinforced with teel plate 7

19 Table 4. G, ψ and ω reult for andwich DCB-UBM pecimen with PVC H45 foam core reinforced with teel layer Moment Ratio, MR M [Nmm] M [Nmm] G anal [J/mm] G CSDE [J/mm] ψ [deg] ω[deg] Eq. (33) ψ* [deg] Eq. (3) * calculated uing ω (avg.) = VI. Influence of reinforcement layer thickne A tudy wa conducted to eamine the influence of reinforcement layer thickne on the ω parameter. A moderately dene H00 core with aluminum face heet wa conidered. The face heet and core thicknee were held contant at h f = 6 mm and h c = 30 mm (ee Table for material propertie). The thickne of the teel reinforcement doubler layer (E r = 0 GPa, ν r = 0.3) wa varied from to 6 mm. The mode-miity phae angle ψ, computed uing CSDE method wa ued to obtain ω parameter uing Eq. (33). The moment ratio (MR) wa varied between -0.5 to 0.5. Reult for the phae angle and ω parameter are provided in Table 5. For h r = mm, the maimum deviation in ω i ±.7%. Similar reult are found for the other reinforcement thickne. Thi confirm that the reult of ω for each reinforcement thickne concurred with the load-independent ω hypothei. The reult how, however, that ω depend on the thickne of the reinforcement layer. The difference between ω for thin (64.9 ) and thick (58.30 ) teel reinforcement layer i 0.%. Average ω value obtained for each reinforcement thickne and MR value are lited in Table 5. Thee value are ued to compute the phae angle ψ, here denoted ψ* from Eq. (3). Value of ψ* in Table 5 may be compared againt the modemiity phae angle (ψ) obtained uing FEA. For mot cae, ψ* agree with ψ obtained uing FEA within 3%. A maimum deviation of 9.% i oberved for MR = and h r = 6 mm. 8

20 Table 5. ω parameter for a H00 andwich pecimen with varying reinforcement thicknee H00 core (30 MPa), hc = 30 mm, hf = 6 mm, hr = - 6 mm Moment Ratio, MR M [N.mm] M [N.mm] hr = mm Ganal [J/mm] ψ[deg] ω[deg] ω (avg) = 64. ψ *[deg] hr = mm Ganal [J/mm] ψ[deg] ω[deg] ω (avg) = 6.7 ψ*[deg] hr = 3 mm Ganal [J/mm] ψ [deg] ω[deg] ω3 (avg) = 6.4 ψ*[deg] hr = 4 mm Ganal [J/mm] ψ[deg] ω[deg] ω4 (avg) = 60. ψ*[deg] hr = 5 mm Ganal [J/mm] ψ[deg] ω[deg] ω5 (avg) = 58.8 ψ*[deg] hr = 6 mm Ganal [J/mm] ψ[deg] ω[deg] ω6 (avg) = 58.3 ψ*[deg] *Phae angle (ψ) computed with an average ω = 60.9 uing Eq. (3) 9

21 VI. (A). Parametric tudy on influence of reinforcement layer thickne (hr) In order to further eamine the dependence of the ω parameter on the reinforcement layer thickne, the tudy i etended to other foam core andwich configuration: E-gla/H45, E-gla/H00, E-gla/H50, Al/H45 and Al/H50. The E-gla face laminate conidered are quai-iotropic (ee Table for material propertie). Face thicknee of 6 and mm were conidered for the gla fiber and aluminum face. A before, the teel reinforcement layer (E r = 0 GPa, ν r = 0.3) thickne i varied from to 6 mm. The ω parameter i calculated from Eq. (33). An average from two MR value (MR = and +0.50) wa ued to determine ω. Fig. 8 how a plot of ω v thickne of the teel reinforcement layer. It i noticed that when tiff aluminum face heet are combined with a oft core (H45), the omega variation acro the range of reinforcement layer thicknee, i 9%. However, for tiffer core the variation i below 5%. For andwich pecimen with E-gla face heet and H45 core, a deviation of 7.% in ω acro h r i oberved. The deviation in ω acro h r for andwich pecimen with E-gla face heet and tiffer H50 core i below.8%. The trend in ω v h r are quantified uing a polynomial curve fit to the data. The curve fitting parameter are provided in Table 6. Figure 8. Omega parameter variation for typical andwich pecimen acro reinforcement thickne 0

22 Table 6. Curve fitting parameter for ω v hr plot for E-gla/PVC core and aluminum/pvc core andwich ytem ω = ς + ς Aluminum/PVC Core, h Al/H45 Al/H00 Al/H50 E-gla/PVC Core, E-gla/H45 E-gla/H00 E-gla/H50 ς = -.649, ς = 6.33 ς = -.5, ς = ς = -.58, ς = 65. ω = η h + η h + η 3 η = -0., η =.34, η 3 = 76. η = -0.40, η =.7, η 3 = 70.5 η = , η =.90, η 3 = 65.4 VII. Concluion Cloed form epreion for energy releae rate and mode-miity phae angle for a reinforced DCB-UBM andwich pecimen were derived uing a uperpoition cheme, the J-integral and laminate beam theory. The phae angle wa epreed in term of a load independent calar parameter ω. Finite-element analyi wa ued to determine energy releae rate and mode-miity phae angle for the variou andwich ytem analyed. It wa found that the ω value remained practically independent of the loading configuration for a fied reinforcement thickne. The value of ω varie weakly with reinforcement thickne and the dependence i epreed by curve fitting for typical andwich pecimen. The cloed form epreion derived in thi paper can be ued for fracture analyi of variou andwich ytem with thin face heet requiring reinforcement layer. APPENDIX A. J-integral calculation Each beam (# and #) i analyzed eparately, Fig. A and A. The J integral i calculated from the tre σ in each layer along the path: Γ Γ 3 and Γ 3 - Γ 0, ee Fig. 5. The tre, σ, in each layer due to moment and force wa calculated and ubtituted in Eq. (5). All equation are epreed in term of the elatic modulu for ply k in the -direction, E. A eplained in ection II, for plane train, Ek= Ek/( ν k ) while for plane tre, Ek= Ek. k

23 Debonded beam (Beam #): Figure A. Force and moment acting on beam # Figure A how the beam coniting of upper face heet and tiffener layer acted upon by a force and moment according to the uperpoition analyi. The force P, and moment M d act on the neutral ai (ee Fig. A). The location of the neutral ai i given by the ratio of etenion-bending coupling tiffne and etenional tiffne ( y NA = BA). The tre σ in each layer i epreed a follow: PEr MdEr + ( y y ); [ Re inforcement] NA y y y ( Eh) B d D d d A d σ = PE f M de f + ( y y ); NA y0 y y [ UpperFaceheet] ( Eh) B d D d d A d (Aa) (A) (Ab) where ( Eh) d= Erhr+ E f h f. The J-integral i calculated for debonded beam by ubtituting Eq. (A) in Eq. (5) along path (Γ 9 and Γ 0) to obtain: y PEr MdE r JΓ9 = ( y ed ). dy E r y ( Eh) H d d ( Eh) d ( ) PEh h 3h h hh = r r M der r f r PM der f r ehh d f r eh d r eh d r H Eh H d d d (Aa)

24 y PE f M de f JΓ 0 = ( y ed ). dy E + f y ( Eh) H 0 d d ( Eh) d ( ) P E h M E h h PM E h h = f f d f f f 3h d f r f r ehh d f r eh d f eh d f H Eh H d d d (Ab) where B H D d d = d A d and e d = y (neutral ai in Fig. A). NA The J-integral contribution from the debonded beam become: J J J Debonded = Γ9 + Γ 0 (A3) P M PM J = L + L + L (A4) Debonded d d 3 H Eh H d d ( Eh) d ( ) d where L = ( Eh r r + Efhf ) (A5a) Er hr 3 Ef hf 3 L = hf hr ehh d f r eh d r hf hr ehh d f r eh d f (A5b) hh f r hh f r L3 = Er eh d r Ef + eh d f (A5c) 3

25 Subtrate beam (Beam #): Analyi imilar to the one above for the debonded beam i conducted here. The layer of the ubtrate beam are the lower reinforcement layer, bottom face heet and core (ee Fig. 0). J i evaluated along path (Γ Γ 3) in Fig. 5. The tre σ due to P and M * (Fig. A) can be epreed a: * PEc M Ec NA 3 ( Eh) B D A ( Eh) D A * PEr M Er ( Eh) B D A ( y y ); y y y [ Core] * PE f M E f σ = NA B ( y y ); y y y [ LowerFaceheet] ( y y ); y y y [ Re inforcment] NA 0 (A6a) (A6b) (A6) (A6c) where ( Eh) = Eh r r + E f h f + Eh c c. The location of neutral ai (y NA = B/A) for the ubtrate beam (#) i hown in Fig. A. Figure A. Load acting on ubtrate beam # Subtituting Eq. (A6) in (5) for each path (Γ Γ 3): y3 * PEc M E c JΓ = ( y e ). dy E c y ( Eh) H ( ) P E h M E h 3hh h 3h PM E h h + h = ( Eh ) * * c c c c 3hr r f c f c c f r eh c( hf hr) eh c eh r H ( Eh) H (A7a) 4

26 y * PE f M E f JΓ = ( y e ). dy E f y ( Eh) H ( Eh) ( ) P E h M E h h PM E h h h = * * f f f f 3hr 3hh r c 3hc f f f r c eh f ( hr hc) eh f eh f H ( Eh) H (A7b) y * PEr M E r JΓ3 = ( y e ). dy E r y ( Eh) H 0 ( ) P E h M E h h 3h PM E h h h = ( Eh ) * * r r r r r 3hc 3 f r r c f hh c f eh r ( hc hf ) eh r eh r H ( Eh) H (A7c) J for the ubtrate beam i obtained by ummation of contribution from the individual layer: J = J + J + J (A8) Subtrate Γ Γ Γ3 Subtitution yield: * * P M PM J = V + V V (A9) ( Eh) ( ) Subtrate 3 H Eh H where V = ( Eh c c + Efhf + Eh r r) (A0a) Ec hc 3 3 V =.5 ( ) hr + hrhf + hf + hc ehc hf + hr + ehc Ef hf 3 3 hr.5hh r c + hf + hc eh f ( hr hc) + eh f Er hr 3 3 hr.5hh f c hf hc eh r ( hc hf ) eh r ( + ) h ( h h ) ( ) hc hf h r h f r c r hc h f V3 = E c eh c + Ef eh f + E r eh r (A0b) (A0c) 5

27 where H B = D A ; e yna = (neutral ai in Fig. A). Now, M* can be epreed in term of M d a, * M Md P = + (ee Eq. ()). Subtitution for M* in Eq. (A9) yield, ( M + P ) P( M + P ) P J V V V d d Subtrate = ( Eh) H ( Eh) H M P M P M P = P P V d V V d d V V 3 V 3 H H H Eh H Eh H ( Eh) ( ) ( ) (A) The J integral for both ubtrate and debonded beam are ummed to obtain the total J a: J = G = J + J = J + J + J + J + J (A) Debonded Subtrate Subtituting Eq. (A4) and (A) in (A): L V V V3 L V G = P M d + + ( Eh) ( Eh) H ( Eh) H H d H d V L3 V 3 Md P H ( Eh) H d ( Eh) H d (A3) Appendi B. Homogenization of laminate face heet Several andwich panel employ multi-directional compoite laminate, Fig. B. The analyi preented here aume iotropic contituent where E f refer to Young modulu of an iotropic material. To ue the analyi preented here for andwich pecimen with compoite laminate face heet, a homogenized modulu hould be appropriate. 6

28 Figure B. Schematic of typical face laminate. The homogenized modulu may be computed uing laminated plate theory [6]. For an element of a laminate, the tre reultant may be epreed a: 0 N A B ε = M B D κ (B) where ε o are the mid-urface train and κ are the mid-urface curvature. A, B and D repreent the etenional, coupling and bending tiffne matrice of the laminate. It hould be noted that ymmetrical laminate are conidered for thi evaluation, hence B = 0. The 6 6 matri in Eq. (B) may be inverted to obtain the compliance matri: ε a a a N ε y a a a N y 0 γ y a6 a6 a N y = κ d d d M κ d d d M 6 y 6 y κ y d M 6 d6 d66 y (B) By ubjecting the laminate trip to an aial load N only, the etenion train become: ε = a N (B3) 0 The effective etenional tiffne can be written a: 7

29 E e σ N = = = (B4) ε ha N ha 0 Similarly, by applying only moment about the -ai, the curvature can be epreed in term of moment, M a: M κ = = Md (B5) E f With I = wh 3 / and M = M/w, thi analyi provide the effective fleural modulu: f E = 3 (B6) d h The average value of e E and f E may be ued to replace E in the analyi. Acknowledgment The financial upport from the Danih Centre for Compoite Structure and Material (DCCSM) funded by the Danih Innovation Foundation (Grant: ) i gratefully acknowledged. Furthermore, the financial upport of Otto Mønted Foundation (Grant: ) to the firt author for hi viit to Florida Atlantic Univerity i gratefully acknowledged. The third author received upport from the National Intitute of Aeropace (NIA) and the Office of Naval Reearch, US Navy (ONR). The NIA and ONR program manager, Dr. Ronald Krueger, and Dr. Yapa Rajapake howed keen interet in thi project and are gratefully acknowledged. Reference [] L. A. Carlon, L. S. Sendlein, and S. L. Merry, Characterization of Face Sheet/Core Shear Fracture of Compoite Sandwich Beam, J. Compo. Mater., vol. 5, no., pp. 0 6, 99. [] S. Praad and L. A. Carlon, Debonding and crack kinking in foam core andwich beam II. Eperimental 8

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