Geometrically non-linear multi-layer beam with interconnection allowing for mixed-mode delamination

Transcription

1 Geometrcally non-lnear mult-layer beam wth nterconnecton allowng for mxed-mode delamnaton Leo Škeca, Gordan Jelenć a, a Faculty of Cvl Engneerng, Unversty of Rjeka, Radmle Matejčć 3, Rjeka, Croata Abstract In ths work we assess the extent to whch a beam model s sutable for the fnte-element analyss of composte structures undergong a large-dsplacement delamnaton process. We lay down the necessary theory needed for the geometrcally non-lnear analyss usng Ressner s beam theory for the layers to be appled to layered structures nvolvng dual-mode damage-type b-lnear consttutve law for the nterconnectons, run a number of representatve examples and compare the results to those obtaned usng a geometrcally lnear analyss. The formulaton s gven n a general form where the number of layers and nodes of the beam fnte elements s arbtrary. To solve numercal problems, the equlbrum of whch s necessarly more complex and demandng to satsfy than n the geometrcally lnear case, the standard cylndrcal arc-length procedure s used only when there s no damage at the nterconnecton. When damage at the nterconnecton occurs, the standard arc-length method has been modfed so that n each load step the converged soluton s requred to result n an ncrease n the total damage of the system. It s concluded that the geometrcally lnear formulatons can be used wth satsfactory accuracy only n lmted number of cases where dsplacements and rotatons reman small. Keywords: mult-layered structure, mxed-mode delamnaton, blnear damage law, non-lnear analyss, mult-layered beam fnte element Correspondng author Emal address: gordan.jelencunr.hr (Gordan Jelenć) Preprnt submtted to Engneerng Fracture Mechancs October 26, 2016

2 1. Introducton Structures composed of multple layers can be found n many areas of engneerng as well as n nature. The most prevalent falure mechansm of such structures s delamnaton n whch the connecton between the layers s beng progressvely damaged due to crackng and s eventually completely lost. Obvously, ths falure mechansm s very complex for a varety of reasons. To start wth, t exhbts overall structural softenng upon reachng a partcular strength of the nterconnecton [1] and n order to assess ths strength t becomes necessary to nvoke the fundamental energy prncples from the theory of fracture mechancs [2]. The actual softenng may be descrbed exponentally, as n the lnear fracture mechancs (see e.g. [3]) or as a lnear or mult-lnear curve, often used n numercal analyses. The global manfestaton of post-crtcal softenng may often become apparent n consderably larger overall dsplacements compared to those n the pre-crtcal range necesstatng a geometrcally non-lnear structural analyss. In addton, nstead of consderng the delamnaton stress at the crack tp as nfnte, whch follows from the prncples of lnear fracture mechancs [4], n real practcal problems t becomes necessary to recognse that the fracturng process s governed by a fnte stress dstrbuton over a small regon around the crack tp, the so-called process zone n Barenblatt s cohesve zone models (CZM) [5]. The cohesve zone models enable the stresses to straddle a narrow crack and descrbe a varety of physcal phenomena rather well, from generaton and localsaton of a prncpal crack [6, 7, 8] to aggregate nterlockng n concrete structures [9]. Also, a crack between two layers may occur for dfferent reasons leadng to the so-called Mode I, II or III openngs (normal to the crack surface, or tangental to t due to slppage or tearng) [1]. Obvously, these may not be consdered separately snce even a lmted damage n a partcular mode always comes as a consequence of some underlyng physcal re-arrangement of partcle bonds on a suffcently small scale whch necessarly reduces also the strengths 2

3 n the other modes. It thus becomes necessary to defne a certan scalar measure of overall damage (see e.g. [10]), whch nvolves contrbuton from all possble modes and governs the phenomenon of damage-nduced strength reducton n all the modes. When modellng engneerng problems we are naturally led by the demands of () accuracy and () computatonal effcence, whch need to be met to wthn a prescrbed measure and n some sense optmsed. For the class of problems analysed here, n our prevous work [11] t has been shown that usng beam fnte elements nstead of 2D solds for planar geometrcally lnear delamnaton gves results of comparable accuracy usng sgnfcantly less degrees of freedom. Such elements do not appear to be as wde-spread n ths type of analyss as the solds, and t s thus argued that they should be consdered as a vald alternatve n a varety of stuatons, ncludng mxed-mode delamnaton. The effcency of mult-layer beam fnte elements n comparson wth commonly used 2D solds has been shown also n authors prevous work [12] where the connecton between the layers was assumed to be absolutely rgd (see also [13] and [14]). In ths work we attempt to assess the extent to whch the beam model and, more generally the geometrcally lnear set-up tself, are applcable to the analyss of the composte structures undergong a delamnaton process. Not unexpectedly, such structures are usually desgned to take advantage of the partcular propertes of the materals formng the composte wthout beng damaged n the operatonal state. However, f we want to trace the post-crtcal equlbrum path after the process of delamnaton has ntated, possbly all the way up to full rupture, we have to recognse that the rato between the dsplacement and the loadng magntudes may ncrease consderably. There also exst such delamnaton phenomena, e.g. peelng, n whch the dsplacements are of the order of magntude of the geometry of the problem analysed. In such stuatons, obvously, geometrcally lnear analyss may not return the results representatve of the real behavour of the problem analysed. Gven the complexty of the delamnaton process, t s not always possble to tell n advance f the geometrcally non-lnear effects may not n fact become consder- 3

4 65 able even for deformaton magntudes whch we may be tempted to ntutvelly classfy as small. In ths work we wll lay down the necessary theory needed for the geometrcally non-lnear analyss usng Ressner s beam theory for the layers to be appled to layered structures nvolvng dual-mode damage-type b-lnear consttutve law for the nterconnectons. In order to assess the need for the geometrcally nonlnear analyss presented, we wll run a number of representatve examples and compare the results to those obtaned usng a geometrcally lnear analyss Problem descrpton Geometry of deformaton of a mult-layer beam s descrbed n [11] and here we reproduce t for reference. An ntally straght mult-layer beam composed of n layers and n 1 nterconnectons s consdered. An arbtrary nterconnecton α s placed between layers and + 1. Fgure 1: Poston of a segment of a mult-layer beam wth nterconnecton n the materal co-ordnate system Materal co-ordnate system of each layer s defned by an orthonormal trad of vectors E 1,, E 2,, E 3,, wth axes X 1,, X 2,, X 3, (see Fg. 1). The axes X 1, are parallel wth the layer s edges and mutually (E 1 = E 1, and X 1 = X 1, ) concde wth the reference axes of each layer. The poston of a reference axs over the layer s heght a 0, h may be chosen arbtrarly, where h s the layer s heght. However, n [11] t was shown that the poston of the reference 4

5 axs may nfluence the numercal results. The cross-sectons of all layers have a common vertcal prncpal axs X 2 defned by a base vector E 2 = E 2, (a condton for a planar deformaton). Note that, accordng to Fg. 1, the coordnate X 2, s dfferent for each layer. Axes X 3, are mutually parallel (X 3 = X 3, and E 3 = E 3, ), but they do not necessarly concde wth the horzontal prncpal axes of the layers cross-sectons. The frst and the second moment of area of the layer s cross-secton wth respect to axs X 3, are defned as S = X 2, da, I = (X 2, ) 2 da, (1) A A where A s the area of the cross-secton of the layer. In our model t s assumed that the thckness of the nterconnecton s very small compared to the layers thcknesses,.e. the geometry of an nterconnecton α s completely defned by ts heght and wdth, denoted as s α and b α, respectvely. The drecton of reference axes of all the layers n the ntal undeformed state s defned by the unt vector t 01 whch closes an angle ψ wth respect to the axs defned by the base vector e 1 of the spatal co-ordnate system (see Fg. 2). Vector t 02 defnes the orentaton of layers cross-sectons whch are orthogonal to the layers reference axes. Thus, the followng relatonshp can be establshed: t 0j = Λ 0 e j = cos ψ sn ψ sn ψ e j, where j = 1, 2. (2) cos ψ Accordng to Fg. 2, the poston of a materal pont n the layer, T (X 1, X 2, ) n the undeformed state s defned by the vector x 0, (X 1, X 2, ) = r 0, (X 1 ) + X 2, t 02, (3) 100 where r 0, (X 1 ) s the poston of the ntersecton of the plane of the cross- secton contanng the pont T and the reference axs of the layer n the undeformed state. The cross-sectons of the layers reman planar but not necessarly 5

6 Fgure 2: Poston of a layer of the composte beam n the spatal co-ordnate systam n the undeformed and a deformed state 105 orthogonal to ther reference axes durng the deformaton of the beam (Tmoshenko beam theory wth the Bernoull hypothess) and the materal base vector E 3 remans orthogonal to the plane spanned by the spatal base e 1 and e 2. Orentaton of the cross-secton of layer n the deformed state s defned by the base vectors t,j = cos(ψ + θ ) sn(ψ + θ ) e j = Λ e j, where j = 1, 2. (4) sn(ψ + θ ) cos(ψ + θ ) 110 Rotaton of the cross-secton of layer s denoted as θ and t s entrely dependent on X 1, thus θ = θ (X 1 ). The poston of materal pont T n the deformed state can be thus expressed as x (X 1, X 2, ) = r (X 1 ) + X 2, t,2 (X 1 ), (5) 6

7 where r (X 1 ) s the poston of the ntersecton of the plane of the crosssecton contanng the pont T and the reference axs of layer n the deformed state. Thus, the dsplacement between the deformed and the undeformed reference axs can be defned for each layer as u (X 1 ) = r (X 1 ) r 0, (X 1 ). (6) 3. Governng equatons 115 The frst group of governng equatons defnes how the layers and the nterconnectons are assembled nto a mult-layer beam. The knematc, consttutve and equlbrum equatons are then defned for the layers, as well as for the nterconnectons Assembly equatons A segment of the mult-layer beam s shown n Fg. 3 n ts undeformed and deformed state. Fgure 3: Undeformed and deformed state of a segment of a mult-layer beam wth nterconnecton 7

8 120 To defne relatve dsplacement between layers and + 1 at nterconnecton α t s necessary to defne the dsplacements at the top and the bottom of the nterconnecton as u T,α = u +1 + (t 02 t +1,2 )a +1, (7) u B,α = u + (t,2 t 02 )(h a ), (8) Vector z α, whch represents a drected stretched thckness of nterconnecton α, can be expressed accordng to Fg. 3 usng (7) and (8) as z α = s α t 02 + u T,α u B,α = = u +1 u + a +1 (t 02 t +1,2 ) + (h a )(t 02 t,2 ) + s α t 02. (9) 3.2. Governng equatons for layers 125 Governng equatons for each layer consst of knematc, consttutve and equlbrum equatons Knematc equatons The knematc equatons are the exact non-lnear equatons accordng to Ressner s beam theory [15] and notaton ntroduced by Smo & Vu-Quoc [16]: γ = ɛ γ = ΛT r E 1 = Λ T (t 01 + u ) E 1, (10) κ = θ, (11) 130 where ɛ, γ, κ are the axal, shear and rotatonal stran (nfntesmal change of the cross-sectonal rotaton) at the reference axs of layer, respectvely. Snce these quanttes are functons of only X 1, the dfferentaton wth respect to X 1 s ntroduced and denoted as ( ). 8

9 Consttutve equatons 135 In ths work the layers are assumed do be made of lnear elastc materal wth E and G as Young s and shear modul of each layer s materal. The axal stran of a fbre at the dstance X 2, from the reference axs of the layer can be computed as ε = ε (X 1, X 2, ) = ɛ (X 1 ) X 2, κ (X 1 ), (12) 140 where ɛ (X 1 ) s the axal stran of a fbre at the layer s reference axs. For the lnear elastc materal the normal stress follows as σ = σ (X 1, X 2, ) = E ε (X 1, X 2, ), (13) whle the shear stress s assumed to be constant over the cross secton (T = G γ ). From (12) and (13), n contrast, t can be clearly noted that the dstrbuton of normal stresses over the layer s heght s lnear. The stress resultants then read N = σ da, (14) A T = G k A γ, (15) M = X 2, σ da, (16) A 145 where N, T, M are the axal force, shear force and bendng moment wth respect to the reference axs of layer, respectvely. The shear correcton coeffcent for layer comes as a consequence of the assumpton of constant shear over the cross secton ntroduced earler and s denoted as k [17]. Substtutng (12) and (13) n (14)-(16) we fnally obtan N M = C γ κ, (17) 9

10 150 where N T = N T T, γ T = ε γ T and E A 0 E S C = 0 G k A 0 E S 0 E I (18) s the consttutve matrx of layer Equlbrum equatons Contnuous form. Equlbrum equatons for layer are derved from the prncple of vrtual work, where the total vrtual work of the layer V L dfference between the vrtual work of nternal forces V nt of external forces V ext actng on layer. Ths can be wrtten as s the and the vrtual work V L V nt V ext = L 0 (γ N + κ M ) dx 1 L 0 ( u f + θ w ) dx1 u (0) F,0 θ (0)W,0 u (L) F,L θ (L)W,L, (19) 160 where γ and κ are the vrtual strans, whle u and θ denote the vrtual dsplacements and rotatons, whch are all functons of X 1. The dstrbuted external forces and moments over the beam s length are denoted as f and w, whle the correspondng pont loads concentrated on the beam ends are denoted as F,0, W,0, F,L, W,L. The vrtual strans are the lnear parts of the strans n (10) and (11) wth respect to the (vrtual) dsplacements and rotatons and can be expressed as γ κ = ΛT 0 I 2 d dx 1 ˆt 3 (t 01 + u ) 0 T 1 0 T d dx 1 u θ = L (D p ), (20) where 0 = {0 0} T, I 2 s a 2 2 dentty matrx and ˆt 3 =

11 165 Expresson (19) now becomes V L = L 0 (D p ) T L T N M pt f w dx 1 p T (0) F,0 W,0 pt (L) F,L W,L (21) Dscrete form. The resultng expresson s hghly non-lnear n terms of the basc unknown functons (u and θ ) and eventually leads to equlbrum whch cannot be found n a closed form. Thus, the shape of the vrtual (test) functons (u and θ ) s chosen n advance assumng that for a fnte number of nodes N on a fnte element the vrtual dsplacements and rotatons are known at the nodes (u,j and θ,j, j {1, N}) and nterpolated between them as p. N = Ψ j (X 1 ) j=1 u,j θ,j N = Ψ j (X 1 )p,j, (22) where Ψ j s a 3 3 matrx of nterpolaton functons. If we further ntroduce the nodal global vector of vrtual unknown parameters p G,j = p 1,j p 2,j... p n,j T for all the layers n the fnte element, we can wrte j=1 175 p = N j=1 [δ 1 Ψ j δ 2 Ψ j... δ n Ψ j ] p G,j = where δ j s the Kronecker delta defned as 1 f = j, δ j = 0 otherwse. At ths pont, expresson (21) can be wrtten as N P,j p G,j, (23) j=1 (24) { L N V L = p T G,j (D P,j ) T L T j=1 0 P T F,0,j(0) P T,j(L) W,0 N M P T f,j dx w 1 } N = p T G,jg L,j, (25) F,L W,L j=1 11

12 180 where g L,j (the term wthn the braces of (25)) s the nodal vector of resdual forces for the layer whch wll be later ntroduced to the global equlbrum equaton of the mult-layer beam wth nterconnecton. It should be noted that D and L now depend on the current confguraton (see (20)), n contrast to the procedure gven n [11]. Ths s where the geometrc non-lnearty of the layers deformaton s accounted for Governng equatons for nterconnectons 185 Each nterconnecton allows for delamnaton n sngle modes (I and II), as well as for the mxed-mode delamnaton. Non-lnear consttutve law wth the embedded cohesve zone model (CZM) [10] s assumed for drectons correspondng to modes I and II. Mxed-mode delamnaton s determned by combnng the nfluence of ndvdual modes. The governng equatons for each nterconnecton agan consst of knematc, consttutve and equlbrum equatons Knematc equatons In order to determne the delamnaton n ndvdual modes, frst we have to defne the drectons correspondng to modes I and II. In case of large dsplacements and rotatons defnng tangental and normal separaton at the nterconnecton s not unque and may be defned n a number of ways. The lne along whch tangental separaton between layers (mode II delamnaton) occurs lays somewhere between the tangent to the reference axes of layers and + 1 and can be defned by the angle θ m α = ζ(ψ + θ ) + (1 ζ)(ψ + θ +1 ) = ψ + ζθ + (1 ζ)θ +1, (26) 200 where ζ represents the weght wth a value between 0 and 1. In the present work value ζ = 0.5 has been used n all numercal examples. Analyss of the mpact of coeffcent ζ on the results has been performed n [18] and t has been shown there that varaton of ζ between 0 and 1 has a small nfluence on the results for the examples analysed there. The relatve dsplacements at 12

13 nterconnecton α accordng to (9) can be now decomposed n two drectons correspondng to delamnaton modes I and II and wrtten n a vector as d α = d 1,α d 2,α = Λm α (z α s α t 02 ) = Λ m α (u T,α u B,α ), (27) 205 where Λ m α = cos θm α sn θα m sn θα m. (28) cos θα m Note that ndex 1 corresponds to mode II and ndex 2 to mode I delamnaton Consttutve equatons 210 For an arbtrary nterconnecton α the consttutve law for the drectons correspondng to delamnaton modes I and II s shown n Fg. 4. Ths concept was proposed by Alfano and Crsfeld [10] who used the so-called nterface fnte elements wth embedded cohesve zone model (CZM). Fgure 4: Consttutve law for the nterconnecton: a) mode II (drecton 1) and b) mode I (drecton 2) The current state of damage s defned usng a parameter whch combnes delamnaton n both modes as [( β α (τ d1,α (τ ) η ( ) d2,α (τ ) η ] 1 η ) ) = + 1, (29) d 01,α d 02,α 13

14 215 where d 01,α and d 02,α are the relatve dsplacements at the nterconnecton α at the start of the softenng process n modes II and I (drectons 1 and 2), respectvely, τ s the pseudo-tme varable and s the McCauley bracket [10]. In all numercal examples n the present work η = 2 s used. The maxmum rate of delamnaton n the pseudo-tme hstory 220 β α (τ) = max 0 τ τ β α(τ ) (30) ensures that the damage of the nterconnecton s rreversble. The tractons at the nterconnecton ω α = ω α,1 ω α,2 T are obtaned from the followng consttutve law: where S α d α f β α 0, ω α = [I G α ] S α d α f β α > 0, (31) S α = S 1,α 0, S,α = ω 0,α, G α = g 1,α 0, 0 S 2,α d 0,α 0 sgn(d 2,α ) g 2,α { } d c,α β g,α = mn 1, α = 1, 2, (32) d c,α d 0,α 1 + β α ω 0,α s the contact tracton at the nterconnecton α at the start of the softenng process n drecton, whle d co,α represents the relatve dsplacement correspondng to the total damage of the nterconnecton α n drecton. When β α 0 we have the lnear-elastc behavour of the nterconnecton, whle β α > 0 ndcates the ongong delamnaton and damage process at the nterconnecton. The degree of the damage s defned by the parameter g,α 0, 1], where g,α = 1 means that total damage of the nterconnecton has occurred and the connecton between layers s completely lost (ω α = 0) Equlbrum equatons Contnuous form. Equlbrum equatons for the nterconnecton can be derved from the prncple of vrtual work. It s assumed that no external loads 14

15 235 are appled drectly on the nterconnectons and the vrtual work of nternal forces of the nterconnecton α reads V C α From (27) t can be obtaned L = b α d α ω α dx 1. (33) 0 where d α = Λ m α (z α s α t 02 ) + Λ m α z α (34) Λ m α =θ m ˆt α 3 Λ m α, θ m α ={0 T ζ 0 T p (1 ζ)} p = ϕt p C,α, (35) +1 ] p z α = [ I 2 t,1 (h a ) I 2 t +1,1 a +1 = B αp C,α. Now, (34) becomes p +1 d α = [ˆt 3 Λ m α (z α s α t 02 )ϕ T + Λ m α B α ] pc,α = Y α p C,α. (36) Dscrete form. Usng p C,α = we fnally obtan p p +1. N = P,j N p G,j = R α,j p G,j (37) j=1 P +1,j j=1 245 V C α = N L p T G,jb α j=1 0 (Y α R α,j ) T ω α dx 1 = N p T G,jg C α,j, (38) where g C α,j s the vector of resdual (nternal) forces for nterconnecton α. Agan, t should be noted that Y α n g C α,j s dependent on B α and Λ m α whch, n contrast to [11], now depend on the unknown knematc felds, thus ntroducng geometrc non-lnearty nto the resdual vector for the nterconnecton. j=1 15

16 4. Soluton procedure From (25) and (38) total vrtual work for a mult-layer beam wth n layers and n 1 nterconnectons now reads V T OT = n =1 [ V L + (1 δ n )V C ] = N n p T G,j j=1 =1 [ g L,j + (1 δ n )g C ],j, (39) 250 where the same counter s used for the layers and for the nterconnectons. Snce the total vrtual work of the mult-layer beam must equal zero (V T OT = 0) and p G,j can be chosen arbtrarly, f follows that the nodal vector of resdual forces for the mult-layer beam s wth g j = n =1 [ g L,j + (1 δ n )g C,j] = q nt j q ext j = 0, (40) q nt j = q ext j = n =1 n =1 L 0 L 0 (D P,j ) T L T N L M dx 1 + (1 δ n )b (Y R,j ) T ω dx 1, 0 P T f,j dx 1 + P T F,0,j(0) + P T F,L,j(L), (41) w W,0 W,L 255 actng as the vectors of nternal and external forces of the mult-layer beam, respectvely. Expresson (40) s hghly non-lnear n terms of the basc unknown parameters, thus the soluton should be obtaned numercally. To obtan the tangent stffness matrx, the vector of resdual forces has to be lnearsed. Snce q ext j = 0, only vector of the nternal forces has to be lnearsed ( g j = q nt j ). For the layers, from (20) we have 16

17 D = 0 2 ˆt 3 u, 0 T 0 L T = θ ˆt 3 0 L T 0 T, (42) 0 N M = C γ κ = C L (D p ), 260 where D and L come as a consequence of geometrc non-lnearty of the layers deformaton and 0 2 s 2 2 zero matrx. These terms do not exst n the procedure gven n [11], and wll result n the geometrc stffness matrx. For the nterconnectons, from (36) we have Y = ˆt 3 [ Λ m (z s t 02 ) + Λ m z ] ϕ T + Λ m B + Λ m B, Λ m = θ m ˆt 3 Λ m, z = B p C,, (43) ] B = [0 2 (h a ) θ t,2 0 2 a +1 θ +1 t +1,2, ω = U d = U Y p C,, 265 where Λ m, z and B n Y come as a consequence of geometrc nonlnearty of the nterconnecton deformaton. These terms do not exst n [11] and wll contrbute to the geometrc stffness matrx. Materal non-lnearty, n contrast s treated n the same manner as n [11],.e. 17

18 U = S f β 0, (I G ) S f β > 0 and β < β, (I G ) S J S d v T f β > 0 and β = β, p C, = { p p +1 } T, J = ξ 1, 0 d cj, sgn(1 g j, ), ξ j, =, j = 1, 2, (44) 0 sgn(d 2,α ) ξ 2, d cj, d 0j, (1 + β ) η+1 ( ) η ( ) η 1 v T d1, 1 d2, =. d 1, d 01, d 2, d 02, 270 Here t has to be emphassed that the thrd case (when β > 0 and β = β ) n U s, n contrast to [11], now correctly derved. However, ths error n lnearsaton dd not cause any sgnfcant convergence problem n numercal examples presented n [11]. If we ntroduce n n,3 of resdual forces becomes = LT N M, the layers part of the lnearsed vector g L j = n L =1 0 {P T,jQ + (D P,j ) T [ S + L T C L D ]} p dx 1, (45) 275 where 0 Q = 2 0 n Tˆt, S = 0 2 ˆt 3 n, (46) d 3 0 T 0 dx 1 0 whch obvously depend on the current stress state and thus vanshes n the geometrcally lnear case. Snce u p = θ = N N Ψ k (X 1 ) p,k = P,k p G,k (47) k=1 k=1 18

19 we can fnally obtan where N g L j = K L j,k p G,k, (48) k=1 280 K L j,k = n L =1 0 { } P T,jQ P,k + (D P,j ) T S P,k + (D P,j ) T L T C L (D P,k ) dx 1 (49) s the nodal tangent stffness block-matrx for all layers related to nodes j and k n whch the frst two terms make ts geometrc part, and the last term makes ts materal part. The nterconnectons part of the lnearsed vector of resdual forces reads where n 1 g C j = L b =1 0 R T,j(Ω + Y T U Y ) p C, dx 1, (50) Ω = ϕω Tˆt 3 Y + (Λ m B ) T (ϕω Tˆt 3 ) T + Z, Z = T (h a )(Λ m t,2 ) T ω 0 T 0, (51) T 0 0 T a +1 (Λ m t +1,2 ) T ω 285 whch vansh n the geometrcally lnear case owng to the presence of current nterconnecton tractons ω. Consderng that we fnally obtan N p C, = R,k p G,k, (52) k=1 N g C j = K C j,k p G,k, (53) k=1 19

20 where 290 n 1 K C j,k = L =1 0 (R T,jΩ R,k + R T,jY T U Y R,k )dx 1 (54) s the nodal tangent stffness block-matrx for all nterconnectons related to nodes j and k. The frst term represents ts geometrc part, whle the second term represents ts materal part. Snce g j = g L j + gc j, t follows that the total nodal tangent stffness block-matrx for a mult-layer beam composed of n layers and n 1 nterconnectons related to nodes j and k can be computed as K j,k = K L j,k + K C j,k. (55) On the element level, the vector of resdual forces, the tangent stffness matrx and the vector of ncrements of unknown parameters are assembled as n [11], whle ther global counterparts g, K and p are assembled usng the standard fnte-element procedure [19]. The followng equatons are then repeatedly solved and the unknown parameters, stress resultants and nterconnecton tractons, and the nternal force vectors updated untl satsfyng accuracy s acheved: p = K 1 g, (56) e. usng the Newton-Raphson method. For ntegraton n (41), (49) and (54) we use Gauss quadrature wth N 1 ponts for the layers and Smpson s rule wth 3 ponts for the nterconnectons (for addtonal nformaton about numercal ntegraton for the nterconnectons see [10]). Soluton algorthm has been mplemented wthn the computer package Wolfram Mathematca. For the geometrcally lnear problems presented n [11] the soluton path s obtaned usng the modfed arc-length procedure [10], whch, unlke the standard arc-length procedure, was able to overcome the sharp snapbacks whch eventually occur n the load dsplacement dagram. However, the procedure sometmes returned to prevously obtaned equlbrum states ( back- 20

21 trackng ) whch was overcome by not takng nto account solutons that close a very sharp angle wth the prevously obtaned equlbrum path, and reducng the arc-length. If after a certan number of arc-length reductons the convergence to a satsfactory soluton stll was not obtaned, the arc-length would be then repeatedly ncreased. In general, we assumed that the procedure has converged to a soluton when the norm of the resdual vector s smaller than a pre-defned tolerance,.e. g < tol. In all numercal examples presented n the present work tol = For the geometrcally non-lnear problems, obtanng the equlbrum path usng the same method as for the geometrcally lnear problems has often proven to be more dffcult and sometmes mpossble. Ths has served as the motvaton to propose a new, more robust method, whch s based on the prncple that the total damage of the system n delamnaton problems can only ncrease (n the case of ongong delamnaton) or at least reman unchanged (when delamnaton process has not started, s nterrupted or t s over) wth each new load step. To measure the total delamnaton of the system a total damage parameter g T OT = n 1 N e N s α=1 el=1 s=1 g m α (el, s) (57) s ntroduced, where for an element el and a Smpson s ntegraton pont s, g m α (el, s) takes the mean value between the mode I and mode II damage as gα m (el, s) = g 1,α(el, s) + g 2,α (el, s). (58) 2 Accordng to (32), g,α (el, s), = 1, 2, can take values between 0 and 1, where 0 means no damage, whle 1 represents the total damage. Thus, when total damage s reached n both drectons (total mxed-mode delamnaton) n an element n and Smpson s ntegraton pont s, we have g m α (el, s) = 1. g T OT In ths new, damage-based arc-length procedure, at the start of an analyss = 0 and the soluton algorthm uses the standard arc-length procedure untl the value of g T OT s changed at the begnnng of a load step. Then, snce the damage process has obvously started, a new method of choosng the correct 21

22 root δλ of the arc-length quadratc equaton s used (see [20, 21] for more detal about ths ssue wth standard and modfed arc-length procedure). Ths new method frst checks f al least one root of the quadratc arc-length equaton gves g T OT whch s greater than the g T OT from the prevous load step. If ths s not the case, whch means that there s no ncrease n the total damage of the system, the standard arc-length procedure s used. If there s only one δλ whch gves g T OT greater than the one from the prevous load step, ths δλ s taken to be the correct root as the one whch gves the ncrease of the total damage of the system. In case when both roots of the arc-length quadratc equaton gve solutons whch result n an ncrease n the total damage of the system, the correct root s taken to be the one whch gves smaller norm of the resdual vector (smlar as n the modfed arc-length method). Independent of the method used (standard or the new damage-based arclength procedure) the arc-length sze can be assgned as constant, wth occasonal reductons when the convergence cannot be acheved, or adaptve, defned by the followng equaton: Nt d c() = c( 1), (59) N t ( 1) where c() s the arc-length sze n the current load step, Nt d s the desred number of teratons (whch s defned at the start of an analyss), N t ( 1) s the number of teratons needed to obtan convergence n the prevous load step 1 and c( 1) s the arc-length sze from the prevous load step. If the convergence s not obtaned after a pre-defned maxmum number of teratons Nt max, the load step s repeated wth a reduced arc length c r () = µ 1 c r 1 (), where r s the ordnal number of the load-step repetton, c 0 () := c() and µ 1 < 1 s an arclength reducton factor. If after a pre-defned maxmum number of arc-length reductons N max red there s stll no convergence, the arc-length s set to a new, larger value c j () = µ 2 c j 1 (), where j s the ordnal number of the load-step repetton wth an arc-length ncrease and µ 2 > 1 s an arc-length augmentaton coeffcent, and the procedure wth arc-length reductons s repeated agan for a 22

23 365 pre-defned maxmum number of arc-length ncreases N max aug. In the numercal examples presented n the present paper µ 1 = 0.5 and µ 2 = 1.25 are used. 5. Numercal examples 5.1. Sngle mxed-mode delamnaton 370 In ths example, a so-called end-notch flexure (ENF) specmen s smply supported and loaded wth two forces F 1 = F 2 and F 2 causng the mxedmode delamnaton at the nterconnecton as shown n Fg. 5. Ths numercal test was proposed by M et al. [3] and also analysed n our prevous work [11], where the geometrcally lnear model was used. Fgure 5: End-notch flexure specmen for mxed-mode delamnaton Geometrcal propertes of the specmen are shown n Fg. 5, wth wdth of the beam b = 1 mm ( = 1, 2, the beam s modelled as two-layered) and the notch length a 0 = 30 mm. Snce the materal propertes for the bulk materal n [3] were gven for the orthotropc materal (two Young s modul, one shear modulus and two Posson s coeffcents), n [11], as well as n the present work, only Young s modulus n the longtudnal drecton and the shear modu- lus n the correspondng transverse drecton are used and gven as E = N/m 2 and G = 5200 N/mm 2, = 1, 2, respectvely. The materal propertes for the nterconnecton are ω 0j = 57 N/mm 2, d 0j = 10 7 mm, d cj = 0.14 mm and S j = N/mm 3, j = 1, 2. In ths example, the damage-based arclength procedure presented n Secton 4 s used wth adaptve arc-length and c(0) = 10 3, N d t = 15, N max t = 25, N max red = 10, N max aug = 10. The reference 23

24 axes of both layers are postoned at the plane of the nterconnecton (a 1 = h 1 and a 2 = 0) and 80 equal lnear two-layer beam fnte elements are used Snce n our prevous work [11] we noted that the dsplacements at the end of the equlbrum path reported n [3] consderably exceeded the lmt of small dsplacements and rotatons (dsplacements up to 40% of the total length of the beam), we have found qute nterestng to nvestgate f the use of geometrcally exact formulaton has any sgnfcant nfluence on the results. In fact, the dfferences are very pronounced, as can be notced n Fg. 6, where the dsplacements of the reference axs of the upper layer at the left-hand end obtaned by both the geometrcally lnear and the geometrcally non-lnear formulaton are plotted. It should be notced that, n contrast to the geometrcally lnear formulaton, n the geometrcally exact formulaton the horzontal dsplacement of the free end of the upper layer also exsts and even for the range of loadng values n [3, 11] takes consderable values (cca 20% of the value of the vertcal dsplacement for F 1 = 20 N). Fgure 6: Results for the mxed-mode delamnaton test on the ENF specmen obtaned usng both geometrcally lnear and non-lnear mult-layer beam models 24

25 Fg. 6 shows only a part of the dagram whch we obtan usng the geometrcally exact model for whch F F 2 40N. If we contnue to ncrease the forces F 1 and F 2 further, the system behaves as shown n Fg. 7. It can be observed that the force F 1 reaches the peak at about 136 N, but the vertcal dsplacement never reaches 40 mm shown n Fg. 6 as obtaned for the geometrcally lnear case. In contrast to the geometrcally lnear analyss, n geometrcally non-lnear analyss the dstance between forces F 1 and F 2 reduces as they ncrease. The force F 2 causes bendng of the beam, whle the force F 1 s responsble for the mxed-mode delamnaton at the nterconnecton. It s very mportant to note that n the geometrcally lnear case the bendng moment n the upper layer at the md-span s equal to F 1 L, whereas n the geometrcally non-lnear case t gets progressvely smaller than F 1 L as the loadng ncreases. For ths reason, n the latter case the vertcal dsplacement of the left-hand end of the upper layer necessarly becomes bounded, and so does the mode 1 crack propagaton, too. After the crack reaches the mdspan, a sgnfcant ncrease n the force F 1 s needed to obtan further delamnaton progress and the left-hand sde end of the upper layer s actually decreasng as the mdspan deflecton of the whole beam ncreases. In Fg. 7 can be also noted that the rght-hand sde support sldes and approaches the left-hand sde support as the beam deforms, reducng the span of the beam. It can be concluded that ths example, whch s often reported n the lterature, has to be treated as geometrcally non-lnear, especally when the vertcal dsplacement at the free end of the upper layer exceeds cca 10% of the total beam length (see Fg. 6) Double mxed-mode delamnaton 430 Ths example, frst proposed by Robnson et al. [22] and later nvestgated by Alfano & Crsfled [10, 23], was also reported n our prevous work [11], where we obtaned an excellent agreement of the results usng sgnfcantly less degrees of freedom. The orgnal HTA913 specmen s orgnally made of 24 layers of equal thckness, but snce the connectons between all layers are assumed to be rgd wherever they exst, the structure s modelled as a three-layer beam 25

26 Fgure 7: Behavour of the ENF specmen under large dsplacements and rotatons wth two nterconnectons n the planes where the ntal cracks are postoned and expected to propagate. Fg. 8 shows the geometry of the specmen, where h 1 = mm, h 2 = mm, h 3 = 1.59 mm and wdth b = 20 mm, = 1, 2, 3 (note dfferent length and heght scales n Fg. 8). The reference axes of all layers concde wth ther centrodal axes,.e. a = 0.5h, = 1, 2, 3. The support at the bottom of the left-hand sde keeps the bottom layer fxed (allowng only rotaton), whle the upper layer can slde n only the vertcal drecton under the load F. As t was reported n [11], as the force F ncreases, frst the upper crack propagates and, when the horzontal poston of ts tp reaches the bottom crack, both cracks contnue to propagate smultaneously. It was also notced that, before t starts to propagate, compressve contact tractons occur at the bottom crack. The orthotropc materal propertes for HTA913 gven n [22] are adapted for the beam model as E = GPa, G = 4.5 GPa, ( = 1, 2, 3), whereas for the nterconnecton, accordng to [23] three sets of materal propertes are used 26

27 Fgure 8: Double mxed-mode delamnaton specmen (all dmensons are n mm) (see Table 1). Table 1: Materal propertes of the nterconnecton for the double mxed-mode delamnaton example Case G c1,α G c1,α d 0j,α /d cj,α ω 01,α ω 02,α d 0j,α d cj,α S 1,α S 2,α [N/mm] [N/mm] [MPa] [MPa] [mm] [mm] [N/mm 3 ] [N/mm 3 ] A B C The meshng s ths example s performed as n [11], where two dfferent meshes of quadratc three-layer beam fnte elements are used: mesh 1 for materal case A and mesh 2 for materal cases B and C (see Table 2). Table 2: Fnte-element meshes for the double mxed-mode delamnaton example wth dfferent materal propertes for the nterconnecton Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Total Length [mm] Intal crack α = 2 none α = 1 none none Mesh 1 / No. of FE Mesh 2 / No. of FE In the numercal smulatons for both the geometrcally lnear and nonlnear analyss a constant arc-length c = 5 s used wth occasonal reductons 455 (µ 1 = 0.5), whle the maxmum number of teraton Nt max, the maxmum num- 27

28 ber of arc-length reductons N max red augmentatons N max aug are set to N max t and the maxmum number of arc-length = 50, N max red = 15 and N max aug = 10. To reach the value of the vertcal dsplacement of the left-hand end of the upper layer v 3 (0) = 35 mm (see Fgs. 9-11) we need 193, 326 and 510 load steps for the geometrcally lnear analyss, and 318, 518 and 1426 load steps for the geometr- cally non-lnear analyss, respectvely for the cases A, B and C. Usng the above arc-length, the algorthm jumps over certan parts of the load-dsplacement dagram (ncludng spurous oscllatons) wthout losng convergence, but many tmes the arc-length has had to be reduced (one or more tmes) n order to obtan convergence. Every reducton of the arc-length bascally ncreases the number of the load steps needed to complete the analyss and there are more reductons n the geometrcally non-lnear analyss. From Fgs we can note that spurous oscllatons for cases B and C are larger than for the case A, although the FE meshes used for cases B and C are twce denser than the one used for the case A. Ths means that the materal model of the nterconnecton has more nfluence on the spurous oscllatons than the fnte-element length tself (see Tabs. 1 and 2). Obvously, larger oscllatons demand more load steps because the total length of the equlbrum path s longer (especally for the case C). However, n comparson wth the geometrcally lnear analyss, the spurous oscllatons for each case are smaller The results of the mult-layer beam model for the geometrcally lnear analyss for ths example have been already compared wth the results from the lterature (see [11] for detals) where t was concluded that the the presented model gves comparable accuracy usng sgnfcantly less degrees of freedom. In ths work we further analyse how the ntroducton of geometrcal non-lnearty affects the results n ths example where the dsplacements and rotatons are not small and the problem tself s rather complex (rregular postons of the ntal cracks, non-symmetrc layerng). For all three sets of the materal parameters for the nterconnecton, the dfference between the geometrcally lnear and nonlnear analyss s more pronounced for the parts of the dagram where the bottom 28

29 Fgure 9: Appled force F aganst the vertcal dsplacement of the left-hand sde of the upper layer v 3 (0) for the case A of the materal parameters Fgure 10: Appled force F aganst the vertcal dsplacement of the left-hand sde of the upper layer v 3 (0) for the case B of the materal parameters 490 ntal crack has propagated (the part wth v 3 (0) n the regon from cca. 7 to 35 mm). In addton, only for the case A the equlbrum path after v 3 (0) 19 mm n the geometrcally non-lnear analyss s qute dfferent n comparson not only to that n the geometrcally lnear analyss, but to the experment as well. 29

30 Fgure 11: Appled force F aganst the vertcal dsplacement of the left-hand sde of the upper layer v 3 (0) for the case C of the materal parameters The exact knematc equatons (10) and the new defnton of the drectons for the delamnaton modes I and II (26)-(28), n combnaton wth the materal parameters of the nterconnecton, for the case A result n a behavour where the bottom ntal crack for v 3 (0) between cca. 7 and 19 mm opens and then closes agan, but does not contnue to propagate to the rght-hand sde for larger values of v 3 (0) (only the upper crack contnues to propagate). Usng a denser mesh (Mesh 2 from Table 2) for the case A gves exactly the same behavour (not shown). Obvously, the materal propertes of the nterconnecton gven for the case A do not model the real behavour obtaned by the experment accurately. In contrast to the geometrcally lnear analyss, the overall results of whch have turned out to be largely nsenstve to the varaton of the materal propertes of the nterconnecton descrbed by the cases A, B and C, n the geometrcally non-lnear analyss ths can be asserted only for the varaton of the materal parameters lyng nt the range defned by the cases B and C. We can conclude that n ths example the case B s the most sutable both n geometrcally lnear and non-lnear analyss because of ts rather good agreement wth the expermental results and acceptable sze of the spurous oscllatons (although Mesh 30

31 510 2 s used). In addton, t s useful to note that for the range of values of v 3 (0) between cca. 7 and 35 mm the actual response of the specmen les between the predctons of the lnear and the non-lnear analyss for both case B and case C Bucklng of a double-cantlever beam Allx and Corglano [24] proposed an example where the layers of a doublecantlever beam (DCB) were loaded by two compressonal axal forces whch caused bucklng of the layers and crack propagaton along the nterconnecton (see Fg. 12). The wdth of the specmen was b = b = 1 mm and the materal propertes for the layers read E = N/mm 2 and G = 5700 N/mm 2, = 1, 2. Two perturbatonal forces F 0 = N were appled on each layer to nduce the bucklng n the desred drecton. Snce the geometrcal and materal propertes of the layers, as well as the appled loadng, were symmetrc wth respect to the plane of nterconnecton, pure mode I delamnaton occurred as the ntal cracks began to propagate. For the nterconnecton, the followng materal parameters were gven: S 1 = 10 6 N/mm 3, ω 01 = 50 N/mm 2 and a set of fracture energes G c,1 = {0.2, 0.4, 0.8, 1.6} N/mm wth the correspondng separatons at the complete damage d c,1 = {0.008, 0.016, 0.032, 0.064} mm. No data regardng the FE mesh used n the analyss were gven n [24]. Fgure 12: Specmen for bucklng n a DCB The example s here run usng the proposed algorthm, where frst the forces F 0 are appled to obtan the ntal deformed confguraton and then the damagebased modfed arc-length procedure s appled (as presented n Secton 4), where only the load F s varable, whle F 0 s kept constant. After the frst load step, 31

32 where arc-length s c(1) = 0.001, the arc-length s changed n each load step ac- cordng to (59) wth N d t = 15, N max t = 25, N max red = 10, N max aug = 10, µ 1 = 0.75, µ 2 = 1.5 and a constrant c() 2, > lnear two-layer beam fnte elements are used n the analyss, where the reference axes of each layer concde wth ther centrodal axes (a = 0.5h, = 1, 2) n order to avod eccentrcty of the axal loads The present model, where the exact geometrcal non-lnearty s accounted for, s compared to the model presented by Allx and Corglano [24], where geometrcal non-lnearty s ntroduced n a mult-layer beam model only as an nfluence of transversal dsplacements on axal strans (the second-order theory). In Fg. 13, where the dsplacement of the free end of the upper layer v 2 (L) s plotted aganst the appled force F, for both cases we can observe the same behavour at the begnnng of the process, where we have an almost vertcal lne (very small change of dsplacement wth ncreasng the load F ) before reachng the bucklng force somewhere around 2 N. After the bucklng has started t can be notced that the dsplacement rapdly ncreases wth a slowly ncreasng force F. At a certan pont, dependng on the materal propertes of the nterconnecton, the bucklng deformaton of ndvdual layers damages the nterconnecton, whch s presented by the softenng branches n Fg. 13. There s also a graph presentng how the system would behave f the nterconnecton were completely rgd (G c = ), where t can be noted that the non-physcal dsplacement v 2 (L) 10 mm, obtaned by the model presented n [24], cannot be obtaned usng the geometrcally exact formulaton presented n ths work even for F > 3 N. The dfferences between the two models, as expected, are more pronounced for larger dsplacements, especally n the case where the nterconnecton s completely rgd. 32

33 Fgure 13: Comparson of the results for the bucklng of the DCB specmen 6. Conclusons and future work In ths work we have proposed a geometrcally exact mult-layer beam fnte element formulaton wth nterconnecton allowng for mxed-mode delamnaton. The formulaton s gven n a general form where the number of layers and nodes of the beam fnte elements s arbtrary, as well as the geometrcal parameters for the layers and the nterconnectons, whle the consttutve laws are assumed to be lnear elastc for the layers and a b-lnear mxed-mode damage law for the nterconnecton. In order to solve numercal problems that are, due to the ntroducton of the exact knematc equatons, more complex and numercally demandng, we have proposed a new modfcaton of the arc-length method, where the standard arc-length procedure s used only when there s no damage at the nterconnecton, else n each load step the converged soluton has to result n an ncrease n the total damage of the system. In the numercal examples, we have shown that usng the geometrcally lnear formulaton n cases when the dsplacements of the system are moderate to large can lead to 33

34 sgnfcant dfferences n the results. On the other hand, for examples that can be solved usng only geometrcally non-lnear formulatons (e.g. DCB bucklng n Secton 5.3), we have shown that the proposed geometrcally exact formulaton gves sgnfcant dfferences n the results compared to the second-order beam theory n the post-crtcal regon. Snce the presented geometrcally exact formulaton s more accurate, gves consderable dfferences n the results and s not sgnfcantly computatonally expensve than the other formulatons used n our comparsons, t can be successfully appled to all types of planar delamnaton problems. The geometrcally lnear formulatons, as shown n the examples presented n the present work, can be used wth satsfactory accuracy only n lmted number of cases where dsplacements and rotatons reman small. The presented model wll be further developed by ntroducng rate-dependence nto the nterface s cohesve law (see [25] and [26]). Other developments may nclude the ntroducton of materal non-lnearty (such as plastcty or hyperelastcty) n the layers and applcaton of some hgher-order beam theores whch would allow warpng of the layers cross-sectons and non-lnear stress dstrbuton over layers heght. It s also possble to ntroduce layers wth deformable thckness where strans and stresses transverse to layer s reference axes appear (see [27] for applcaton n mult-layer beams wth rgd nterconnecton). Acknowledgement 595 These results were obtaned wthn the research project No IP (Confguraton-dependent approxmaton n non-lnear fnte-element analyss of structures) fnancally supported by the Croatan Scence Foundaton. We also acknowledge the Unversty of Rjeka fnancal support for ongong research No (Testng of slender spatal beam structures wth emphass on model valdaton). References 600 [1] Z. Bažant, L. Cedoln, Stablty of Structures, Dover,

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