MODELING OF ADHESIVE LAYERS OF LAMINATED PLATES IN DELAMINATION SIMULATION

Transcription

1 MODELING OF ADHESIVE LAYERS OF LAMINATED PLATES IN DELAMINATION SIMULATION G. Shi H. Zhng Deprtment of Mechnics Tinjin University 92 Weijin Rod Tinjin CHINA SUMMARY This pper first briefly reviews e interfce elements bsed on penlty contcts for detion simultions of ted pltes. Then it presents new 3-D interfce element bsed on solid-shell element for detion simultion. Numericl exmples show t e solid-shell interfce element is efficient nd ccurte for detion simultions. Keywords: ted pltes impct dmge detion simultion interfcil lyer modeling solid-shell interfce element INTRODUCTION Fiber reinforced ted pltes nd shells possess high specific modulus nd high specific streng in e reinforcement direction nd ey re widely used in vrious industries such s erospce utomobile ship building construction etc. However e trnsverse streng of ted pltes is very wek especilly t e dhesive lyers which re e resin rich zones t e interr interfces. As result ted pltes will suffer dmges such s mtrix crcking debonding nd inter-ply detions when ey re under e ction of e low-velocity trnsverse impct [1-3]. Among ese impct induced dmges e detion t e interr interfces wi different fiber orienttions is e most dngerous one s it cn severely ffect e streng nd stbility of e dmged tes. The numericl simultion bsed on e finite element meod is powerful tool for e prediction of detions of ted pltes subjected to low velocity impct. The key issue in e detion simultion is e behviour modelling of e interfcil lyers of e ted pltes. In recently yers mny interfce elements were proposed for e modeling of e interfcil lyers [4-9] eier for e 2-D shell element models or for e 3-D element models. These interfce elements including e cohesive zone model re bsed on e penlty contct pproch [2 3 7 mong oers] in which e virtul contct springs nd virtul reltive displcements re used. The virtul contct springs hve given penlty stiffness prmeters to simulte rel connection between two neighbouring lyers before detion initition. Nevereless e determintion of spring stiffness for e penlty stiffness of e interfces is not bsed on e physics of e dhesive lyers but rer rbitrry which leds to some numericl difficulty nd instbility [5-6]. In ese models e onset of e detions is controlled by e streng bsed criteri nd e detion propgtion is governed by mens of e indirect use of frcture

2 mechnics. Vrious mteril models for e virtul springs of penlty contcts hve been proposed so fr [5 10] but ese models hve some difficulties to ccurtely model e inter-ply detions wi e interction of e mixed modes of frctures. Alough e penlty contct-bsed interfce elements re ble to simulte e inter-ply detion resonbly well e ccurcy nd stbility of e simultion results strongly depends on e users good experience on e choice of e virtul spring stiffness [5-6]. Therefore it is desirble to develop new efficient 3-D interfce elements bsed on e rel physics nd geometry of e interfcil lyer properties for e robust detion simultions. The objectives of is pper re two-folded. First is pper briefly reviews e interfce elements used in e numericl predictions of inter-ply detions of composite pltes subjected to low velocity trnsverse impct. Second solid-shell element-bsed [11] interfce element ccounting for e rel geometry nd e mteril properties of e interfcil lyers is proposed for e detion simultion. The preliminry numericl study shows t e new 3-D interfce element presented here is not only ccurte but lso computtionlly efficient compred to e conventionl 3-D interfce elements in e detion simultions of tes pltes. ON THE INTERFACE ELEMENTS BASED ON PENALTY CONTACTS When ted plte is subjected to low velocity trnsverse impcts it will suffer detion dmge t e interr lyers. Becuse of e brupt chnges in stiffness t e dhesive lyers of e wi different fiber orienttions e inter-ply detion would most likely tke plce long e dhesive lyers bonding e wi different fiber orienttions. Therefore e dhesive lyers between e e hving different fiber orienttions cn be modeled s e interfcil lyers of ted pltes. A typicl interfcil lyer between two e is depicted in Fig. 1. The dhesive lyer hs finite ickness but it is much inner n ose of e. h h ( i +1 ) i dhesive lyer ( i +1 ) K h gp K δ 1 i δ 3 () typicl interfcil lyer (b) virtul springs of penlty contct model Figure 1. Typicl interfcil lyer nd penlty contct bsed interfce element In recently yers mny interfce elements bses on e concept of e cohesive zone model hve been proposed for e behvior modeling of e interfcil lyers of ted pltes eier for e 2-D shell element models or for e 3-D element models [2 3 7]. The bsic ide in ese interfce elements is e penlty contct for e cohesive zones in which e stiffness of e virtul contct springs is used s penlty prmeters to simulte rel connection between two neighboring lyers before detion s shown in Fig. 1b. There re two chrcteristic prmeters in e penlty

3 contct bsed interfce models depicted in Fig. 1b one is e stiffness of e virtul spring k nd oer is e virtul ickness of e interfce. However e determintion of spring stiffness nd e ickness of e interfce elements in ese models re not bsed on e physics nd geometry of e dhesive lyers but rer rbitrry. For instnce h gp = 0 is used in some interfce models [ ] but finite vlue in terms of frction of ickness for h gp not e rel ickness of e interfcil lyer i.e. hgp h is defined in oer models [7-9]. The vlue of hgp in some detion simultions cn be dozens of times lrger n e rel ickness of dhesive lyers. For exmple one-fif of e ickness for h gp ws suggested in Ref. 7. Becuse of such nture of e penlty contct bsed interfce models e resulting interfce elements could led to some numericl difficulty nd instbility [5]. The bsic field vribles of interfce elements bsed on e penlty contct re e trctions of e virtul springs p nd e reltive displcements δ between e neighboring e δ cn be treted s e equivlent crck opening displcements for detion of ted composite structures. Before e detions tke plce e reltion between e trctions p nd e reltive displcements δ t n interfce cn be defined s p = D δ (1) where D is e stiffness mtrix of virtul springs for e interfces. The reltive displcements δ nd stiffness mtrix D tke e following forms respectively δ u u k δ = δ 2 = u2 u2 D = (2-3) 0 k2 0 δ 3 u3 u top 3 bottom 0 0 k3 In e penlty stiffness bsed interfce models e detion dmge is supposed to be initited when e mximum trction reches reshold nd en e stiffness of contct springs is degrded. Vrious softening lws of e spring stiffness were proposed so fr [5 10]. The biliner constitutive lw for e virtul spring is e most populr one. A biliner constitutive lw is illustrted in Fig. 2 Fig. 2 is e forcedisplcement curve Fig. 2b shows e equivlent strin energy relese rte of e spring where G i (i = I II III) e re of e tringle OAB is e strin energy relese rte of crck mode i (i = I II III) nd G ic e totl re of e tringle OAC is e corresponding criticl strin energy relese rte. As shown Fig. 2 e streng bsed criteri is used to control e onset of e interfcil dmges nd e indirect use of frcture mechnics is dopted to determine e detion propgtion in which e energy dissipted in e virtul spring is equivlent to e strin energy relese rte in frcture mechnics [ ]. In cse of mixed crck modes of frcture vrious power lws re widely used to pproximte e interction of e ree frcture modes. However e model of power lws is not esy to ccurtely predict e detions ssocited to e mixed crcks modes of frctures since e crck propgtion direction depends not only on e stress regime t e crck tip but lso on e ply orienttion djcent to e crck plne [1]. h gp

4 p i τ i p i mx τ i mx i G i ic G ic δ i δ i () force-displcement curve (b) equivlent energy relese rte curve Figure 2. A biliner constitutive lw for e virtul springs The dvntge of ese interfce models is e simplicity nd e unifiction of dmge initition nd detion propgtion. The disdvntge is t e ccurcy efficiency even stbility of detion simultions strongly depend on e proper choice of e vlue of e spring stiffness. Unfortuntely e determintion of proper vlues of e spring stiffness is totlly bsed on e users expertises s smll vlues of spring stiffness would induce lrge interpenetrtions t e interfces which re incomptible wi e physicl relity nd too lrge vlues of spring stiffness could produce numericl errors relted to e computing ccurcy [5-6]. NEW 3-D INTERFACE ELEMENT BASED ON SOLID-SHELL ELEMENT The numericl simultion results hve shown t e interfce elements bsed on e penlty contct nd e equivlence of dissipted energy to e energy relese rte re cpble of simulting e inter-ply detion resonbly well. Nevereless Becuse e chrcteristic prmeters in ese interfce elements re not e rel properties of e dhesive lyers e ccurcy nd efficiency of e simultion results strongly depends on e users good guess of e virtul spring stiffness. Therefore it is desirble to develop some new interfce elements bsed on e rel physics nd geometry of e interfcil lyer properties for e robust detion simultions. The interfcil lyers re e resin rich zones wi finite ickness nd ey cn be modeled by 3-D elements eoreticlly. However since e ickness of e interfcil lyer is too in compred wi e ickness of e djcent s e mesh density of simultion model using e conventionl 3-D interfce elements is totlly controlled by e suitble spect rtio of e interfce elements representing e in dhesive lyers. Consequently e 3-D element modeling for e interfcil lyers bsed on e conventionl formultion leds to e resulting mesh being too lrge to be computtionlly prcticl [3]. The recently developed solid-shell element [11] is type of specil solid elements. The specil fetures of solid-shell elements is t it hs e 3- D geometry like e conventionl solid elements but it is lso cpble of modeling plte-like structures like plte elements nd giving ccurte results even in e cse of lrge spect rtios of leng to ickness pproching to 100 [11]. Therefore e computtionl inefficiency of e conventionl 3-D interfce element model cn be overcome by e 3-D modeling of interfcil lyers using e solid-shell elements. In ddition e rel mteril properties nd e physicl ickness of e interfcil lyers cn be used directly in e solid-shell interfce elements nd e dmge mechnics cn

5 lso be directly employed rer n e indirect use of e frcture mechnics s e sitution in e populr interfce elements bsed on e penlty contct pproch. The 3- D modeling for e interfce lyers using e solid-shell element is schemticlly shown in Fig.3. The new version of ANSYS dded e solid-shell element into its element librry [12]. The performnce of e solid-shell elements in ANSYS 12.0 is evluted in is study first en 3-D interfce element bsed on e solid-shell element nd e dmge mechnics is proposed. ( i +1 ) h solid-shell element dhesive lyer i 3-D element modeling Figure 3. 3-D interfce element bsed on solid-shell element formultion The dhesive lyer is e resin rich zone wi ickness. Consequently interfce representing e dhesive lyer cn be treted s isotropic mteril wi Young s modulus E nd sher modulus G of e dhesive lyer. Since e interfcil lyers shown in Fig. 3 is modeled s solid medium wi finite ickness h e dmge of e interfcil lyers resulting from impcts cn be modeled directly by e continuum dmge mechnics. A generl norml stress-strin curve of dhesive lyer is shown in d Fig. 4 where E denotes e Young s modulus of e dmged mteril nd ε u is e ultimte strin. h σ B σ s A E d E O ε e ε u ε Figure 4. Mteril modeling of dhesive lyers Since e detion ptterns of interfcil lyers re similr to e ree crck modes of frctures e interfcil detion dmges cn be chrcterized by ree dmge prmeters nd representing e interfce dmge respectively in d3 d1 12 d 2 12 x3 x1 1 x2 e direction e direction of e x plne nd e direction of e x1 x 2 plne s shown in Fig. 5. The dmge prmeters 3 1 nd re e x 2 d d 12 d 2 12

6 functions of mteril properties E G σ s nd τ s of dhesive lyers nd some internl vribles. The progressive dmge process of e inter-ply detion induced by low-velocity impct cn be modeled by proper dmge evolution lw which is widely used in e continuum dmge mechnics [13]. The re of under curve OAB in Fig. 4 is e strin energy density up to e filure. It cn be seen from Fig. 4 t when e ultimte strin ε u is few times lrger n ε e criterion of dmge propgtion. ε cn be used for e u x 3 ( i +1 ) h solid-shell element P P E G σ τ d d d ) i = ( s s x 1 Figure 5. The dmge modeling of interfcil lyer DELAMINATION ANALYSIS USING SOLID-SHELL INTERFACE ELEMENT The performnce of e solid-shell interfce element presented in e previous section is evluted first in is section en it is used for e simultion of e stndrd test of e double cntilevered bem to demonstrte e efficiency nd ccurcy of e solid-shell interfce element. Vlidtion of solid-shell element in ANSYS The deformtion of [0/90/90/0] ted bem under uniform lod q but wi different boundry conditions re nlyzed first to vlidte e ccurcy of e solidshell elements in ted bem nlysis. The mteril properties nd e geometry of e bem re s following: E 1 = 144.8GP E 2 =9.65GP G 12 = G 13 = 4.14GP G 23 =3.45GP ν 12 =0.3 L = mm h=1.0375mm h l = 0. 25mm h = mm where h l = 0. 25mm is e ickness of. The spect rtio of e bem L/h = elements re used long e bem xis direction nd ech is modeled by 4 lyers of 3-D element SOLID45 in ANSYS which result in t e spect rtio α lr of lle leng lle to ickness hle for e elements used for e is α lr = = The h dhesive lyer is modeled by e solid-shell element SOLSH190 in ANSYS but different number of element lyers is used for dhesive lyers to study e influence of le

7 le e spect rtio α r = of element SOLSH190 on e nlysis results where l is h e e interfce element leng nd e nondimensionl mximum deflection e bem is defined s: h e is e interfce element ickness. The w = w mx E 2 bh q 3 L 4 /12 The mximum deflections of e ted bems wi four different boundry conditions re tbulted in Tble 1 where symbols for e boundry conditions re: CC stnds for Clmped - Clmped bem ends CF for Clmped - Free bem ends CS for Clmped- Simply supported bem ends SS for Simply supported - Simply supported bem ends. The solutions given by e composite bem element HQCB-8A [14] which is bsed on higher-order sher deformtion eory nd e ssumed strin meod is lso listed in e tble for comprison. But e dhesive lyer is not considered in e nlysis of HQCB-8A. The results in Tble 1 show t e solid-shell interfce elements deliver good results even when e spect rtio α of e interfce elements is very lrge. e Tble 1. Nondimensionl mximum deflections given by solid-shell interfce elements lyers of interfce elements spect rtio α r CC CF CS SS HQCB-8A [14] Crck propgtion of DCB The double cntilevered bem (DCB) considered here is schemticlly shown in Fig. 6. The geometry of DCB re e leng of e bem l=100mm height h=3mm wid b=1mm initil crck leng 0 =30mm e ickness of dhesive lyer h =0.02mm. The mteril properties of e e [7] re:

8 E 1 =126GP E 2 =E 3 =7.5GP G 12 = G 13 = 4.98GP G 23 =3.28GP ν 12 =ν 13 =0.261 The dhesive lyer re modeled s n isotropic mteril wi mteril properties: E =7.5GP G =3.28GP σ s =57MP ε u =0.015 The DCB shown in Fig. 6 cn be considered s ted bem mde of two e wi n dhesive lyer of 70mm long. 4 lyers of solid elements re used for ech of e composite bem nd one lyer of solid-shell element SOLSH190 re used for e dhesive lyer of e bem s shown in Fig. 7. F h 0 F Figure 6. DCB wi initil crck leng 0 =30mm Three element mesh densities re used for e dhesive lyer nd finer meshes re used round e crck tip. The interfcil element lengs t e cohesive zone re respectively = 0.1mm 0.2mm nd 0.5mm nd e corresponding spect rtios re l l e e α r = = 5 10 nd 25. The finite element mesh wi l = 0.2mm t e crck tip nd h e e e deformtion shpe of DCB t F = 80N (before e pek lod shown in Fig. 8) is depicted in Fig. 7 nd Fig. 7b respectively. () mesh t e crck tip (b) deformtion t F = 80 N Figure 7. Mesh t e crck tip nd simulted crck opening of DCB The force-opening displcement curves obtined by using e solid-shell interfce element re illustrted in Fig. 8. It cn be seen from e figure t e present results re gree well wi e experimentl results [7] when e interfce element leng l e 0. 2mm is is becuse t e crck propgtion is simulted by e mteril filure of e interfce elements nd e propgtion rte of e crck is controlled by e leng of interfce elements used to model e dhesive lyers. l e

9 experi. Figure 8. Simultion results of DCB given by solid-shell interfce elements The results in Tble 1 nd Fig. 8 hve illustrted t e interfce element bsed on e solid-shell element is good meod to model e interfcil lyers of ted structures which is extremely in compred wi ose of neighbouring e but hving finite ickness. CONCLUSIONS Hving reviewed e interfce elements bsed on e penlty contcts n interfce element bsed on e solid-shell element is presented for e interfcil lyer modeling in e detion simultions of tes pltes. The preliminry numericl study shows t e present 3-D interfce element bsed on e solid-shell element hs e following dvntges. 1. The rel geometry of e interfcil lyers of e which re e resin rich zones between e cn be ccurtely modeled by e 3-D geometry of e solid-shell interfce elements; 2. The rel mteril properties of e dhesive lyers cn be used directly for e mteril modeling of e solid-shell interfce element nd e dmge nd e filure of e interfcil lyers cn be chrcterized by dmge mechnics rer n e indirect use of frcture mechnics; 3. The solid-shell interfce element is computtionlly efficient for e modeling of extremely in dhesive lyers in ted pltes since e solid-shell element is lso cpble of giving good results in e cse of lrge spect rtio of element leng to its height. Therefore e solid-shell interfce element is n efficient numericl tool to model e behviour of e in interfcil lyers in e detion simultions of ted structures. ACKNOWLEDGEMENTS The finncil support by e grnts of NSFC nd TSTC-08JCYBJC01100 re nkfully cknowledged.

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