BLADE CUTTING SIMULATION WITH CRACK PROPAGATION
THROUGH THIN-WALLED STRUCTURES VIA SOLID-SHELL
FINITE ELEMENTS IN EXPLICIT DYNAMICS
ALDO GHISI, FEDERICA CONFALONIERI AND UMBERTO PEREGO
Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano Piazza Leonardo da Vinci 32, 20133, Milano (ITALY)
aldo.ghisi@polimi.it, federica.confalonieri@polimi.it, umberto.perego@polimi.it
Key words:Solid Shell Finite Elements, Selective Mass Scaling, Layered Thin-Walled Structures, Explicit Dynamics, Blade Cutting.
Summary. To simulate the crack propagation due to blade cutting of a thin-walled shell structure, we propose a numerical technique based on solid shell finite elements and explicit time integration. The limitation on the critical time step due to the small thickness along the out-of-plane direction is overcome through a selective mass scaling, capable to optimally define the artificial mass coefficient for distorted elements in finite strains: since the selective scaling cuts the undesired, spurious contributions from the highest eigenfreqeuencies, but saves the lowest frequencies associated to the structural response, and since the method preserves the lumped form of the mass matrix, the calculations in the time domain are conveniently speeded up. The interaction of the cutting blade with the cohesive process zone in the crack tip region is accounted for by means of the so-called directional cohesive interface concept. Unlike in previous implementations, through-the-thickness crack propagation is also considered. This is of critical importance in particular in the case of layered shells, where one solid-shell element per layer is used for the discretization in the thickness direction and it is a necessary ingredient for future possible consideration of delamination processes. We show by applying the proposed procedure to the cutting of a thin-walled laminate used for packaging applications that this is a promising tool for the prediction of the structural response of thin-walled structures in the presence of crack propagation induced by blade cutting.
1 INTRODUCTION
In this paper we consider the simulation of the opening process of a thin laminate membrane sealing a carton package by means of cutting blades. This problem is actually challenging from a computational point of view, since the simulation must account for fracture initiation and propagation through a thin-walled laminate. The nonlinear material behaviour of the single layer and the small laminate thickness make recently introduced [1-2] approaches, based on solid-shell elements particularly attractive. Moreover, the dynamic nature of the fracture process and the severe nonlinearity due in particular to the interaction
with the cutting blade suggest to use an explicit dynamics time integration scheme. These issues require a detailed analysis of numerous numerical problems, whose solution is briefly recalled in the following sections. The simulation approach considered here is based on the directional cohesive element concept, first proposed in [14] to simulate blade-induced crack propagation in classical elastic shell elements and later reformulated for application to solid-shell elements [3]. Directional cohesive elements are special interface elements, to be interposed between adjacent shell elements, capable to account for the interaction between the cutting blade and the cohesive process zone. Section 2 briefly recalls the adopted solid-shell finite element formulation, section 3 details the selective mass scaling approach; in section 4 we sketch the ideas, i.e. the directional cohesive elements, at the base of the interaction between the cutting blade and the fracture resistance of the laminate, and in section 5 we show the application of the modelling to an opening laminate in the packaging industry.
2 SOLID SHELL ELEMENT FORMULATION
Solid-shell finite elements, unlike standard, structural shell elements make use of only translational degrees of freedom. Because of the three-dimensional kinematic formulation, the constitutive laws used for continuum, three-dimensional finite elements can be adopted as well, including direct consideration of thickness deformation. Unlike standard finite elements, solid-shell elements can be conveniently used for the discretization of layered thin-walled structures, by simple stacking a number of solid-shell elements, typically one or more per layer, one on top of the other through the shell thickness. In the following we choose the Q1STS finite element proposed by Schwarze and Reese [1-2], an eight-node hexahedron including a reduced integration and hourglass control, recently adapted to explicit dynamics in [6]. Solid-shell elements are affected by numerical problems, such as Poisson, volumetric, curvature and shear locking and by hourglassing, the latter due to reduced integration which is an important ingredient for an efficient explicit time integration scheme. In Schwarze and Reese implementation volumetric and Poisson’s locking are dealt with by the enhanced assumed strain (EAS) approach, while the assumed natural strain (ANS) concept is used for the transverse shear and curvature thickness locking. Hourglassing is controlled by adding suitably constructed artificial stiffness.
3 SELECTIVE MASS SCALING
In explicit structural dynamics the advantage of algorithm simplicity (there are no linear systems to be solved) is balanced by the small allowed stable time step; in the case of the widely diffused central difference scheme, the time step ∆ is dependent on the maximum eigenfrequency max of the finite element mesh according to
∆ 2
max
(1)
in the undamped case [8]. max is in turn conservatively bounded by the maximum
eigenfrequency of the single finite element max:
max max max . (2)
of a she element dynamic pursued down th are seve ideas m integrat where th solid-sh determi inertia, care wh drawbac finite str 3.1 Pri To sum it starts and low Then, th segmen and thes which is
ell, the Cou t traversal t cs. To ove d: the idea i
he highest e eral approac make the sca tion scheme he relative m hell element ning the ti while the r hen a signif ck, we emp rains [4]. inciples of t mmarize our by introdu wer surface ( he coordina nts connectin se items are s related to urant-Friedr time of an ercome this is to artifici eigenfrequen ches to ach aled mass m es. Here we motion betw t is conven ime step an rigid body r ficant comp phasize that the propose selective m ucing the sp (see figure 1 Figure 1: L ates of the c ng the corre 2 e collected i the original ichs-Lewy elastic wav severe lim ially alter th ncies with s hieve this go matrix non e follow the ween the low niently pena nymore. Th rotational in ponent of th t the metho ed selective mass scaling plitting of th 1): Local numberi corner node esponding n n the transf l coordinate condition, l ve, leads to mitation, the he solid-she small or neg oal (see e.g diagonal, a e approach
wer and upp alized so th he approach nertia is mo he motion i od works fo e scaling ap g approach w he element ing adopted fo es at the mid nodes of the formed coor ∆
es via the lin
limiting the o very sma e selective ell element gligible cha g. [13,15,16 an undesira shown in t per shell su hat the sma
h maintains odified; the s a pure rig or regular an pproach we use here nodal coord . or the solid-sh ddle surface upper and l ∆ rdinate vect near transfo e time step all time inc mass scali mass matr anges to the 6,17,18,19]) able outcom the series o rfaces of a 8 allness of th s the transl erefore, it sh gid body ro nd distorted e the notatio dinates acco hell element. e and the co lower surfa 2 or ormation to a fractio crements in ing concept rix in order lowest one ), but most me for expli of papers [4 8-node thre he thicknes lational rig hould be us otation. Bes d finite elem on describe ording to th orner fibers ace, are defin
on of the explicit t can be to scale es. There of these icit time 4,6,9,10], ee-linear, ss is not gid body sed with ides this ments in ed in [4]: he upper (3) s, i.e. the ned: (4a,4b) (5)
(6a,7b)
where is the 12x12 identity matrix. A similar transformation can be followed also for displacements, velocities and, finally, accelerations, which in particular read:
,
∆ . (8a-c)
The principle of virtual work for the motion of an undamped system can be therefore transformed as follows:
,lumped (9)
with equal to the transformed virtual acceleration of the nodes of element e,
the difference between external and internal (transformed) equivalent nodal forces for the single FE, and ,lumped the lumped form of the selectively scaled mass matrix, namely
,lumped (10)
and being 12x12 diagonal matrices collecting the element nodal masses at the
upper and lower surface nodes, respectively, and being the element scaling coefficient. Notice that is applied only to the difference in nodal acceleration between the upper and
lower surface and not to the middle surface nodal accelerations that are responsible for the element translational rigid body motion. In case of a single layer thin-walled structure, this mass matrix evidently does not change the diagonal form of the problem and can be directly inserted in the explicit solver routines written in the transformed variables. In case, instead, of a layered structure, it is necessary to reverse to original coordinates; in fact, after assembly, the nodal masses of adjacent layers belonging to the same multi-layer fiber sum up, thus requiring to solve a small linear system. While a detailed presentation is present in [9], the basic idea can be grasped by considering the three layers (labelled as a, b, c) example in figure 2. For each fiber f in this case the problem to be solved is
(11)
with
and ∑ 1 , ∑ 1 ,
where, for each node, the sum operator has to be interpreted as the assembly over the layer
3.2 Op Unne dynamic depende plateau problem essentia geometr coeffici where shell el approxi readers) element plane di equation bounded ; 4 DIR A sh cohesiv the coh computa The “d energeti element assumpt thicknes ptimal mass ecessary hi cal propert ence of the for increas m has been s ally by exp rical scalin ent turns ou min min lement. Th mate for d ). Moreove t; by using t imensions a n 12 to the t d (details opt 0 vi RECTIONA harp blade ve process z hesive app ationally pr directional” ically consi t interfaces tions of i) n ss, and ii) th s scaling fa gh values ies (e.g. rig
maximum e sing . The solved in [1 ploiting the ng. A good ut to be: , is th he estimate distorted el r, this estim the more re and not on i time step es are given ia a Newton AL COHES cutting a zone during proach wo rohibitive m cohesive istent descr where spec neglecting t he independ Figure 2: ctor of the scal gid body ro eigenfreque erefore, it 10] for paral equivalenc approxima he minimum e is actual ements (in mate is dif estricted est its thickness stimate, we in [4]) by n-Raphson p SIVE ELEM ductile (or g crack prop ould lead mesh, capabl approach [ ription of th ial string el the bending dence of the : Three-layers ling coeffic otational in ency on the is necessar llelepiped e ce between ation of the opt≅ min 0 m in-plane l lly exact f n [4] the ex fferent for e timate, the t s. To pass f follow equa y the soluti procedure. MENTS not purely pagation. In to incorre le to resolve [3,14] is in his interacti lements are g strength o e specific su s laminate. cient can nertia) and mass scalin ry to determ elements an selective s e optimal e n length and for parallel xact solutio each layer time step de from the ma ations 1 and ion of the y brittle) m n this case, ect results e the blade ntended to ion, allowin inserted aft of individua urface work n overly mo it can be ng coefficie mine the op d in [4] for scaling and estimate fo is the thi lepiped ele on is prov of the lam epends only ass scaling c d 2 where element e material can a classical unless an radius of cu provide a ng a crack ter node dup
l layers bec k due to cut odify the s shown [9] ent actually ptimal scali r distorted e d element th or the mass ickness of th ements whi vided for in minate and y on the ele coefficient max is conv eigenvalue n interact w l implement n exterme urvature, we a macrosco to propaga plication. U cause of the tting from t tructural that the shows a ing: this elements, hickness s scaling (12) he solid-ile only nterested for each ment in-stated in veniently problem with the tation of ly fine, ere used. opic but ate along Under the eir small he crack
propaga criterion separati is a geo (figure generati contact the cutt directio Unlik separati solid-sh position displace that the where t between attached The for softenin cohesiv of a lin progress ation mode n is met at ing element metric entit 3a, 3b): the ing a joint i with the lat ting blade) n. (a) Figure 3: (a) ke in the cla ing crack fl hell nodes w n is given a ement disco cohesive fo the subscrip n the cable d in corresp rces are ng law relat ve potential near softeni sive damag (this impl t a given n faces in co ty, capable t e latter, in f in correspon teral surface , and allow ) blade cutting assical cohe flanks, deter with the ca at each time ontinuities orces becom pts “+” an and the bla pondence of calculated ing the ope ℓ, wit ing law, i.e e inside the lies that on node), one m orresponden to detect co fact, modifie ndence of th es of the bl wing for a g through the m and (c) “dir esive approa rmined by able joint (s e, then the are know me: ‖ ‖ ‖ m ‖ ‖ ‖ m nd “-” deno ade and m f the elemen from the c ning tractio th an inte e. the maxi e process zo nly fracture massless ca nce of the el ontact and to es the origi he contact p ade is exclu transmissio (b) mesh, (b) inte rectional” coh ach, these f the directio see figures contact po wn at the en mid ‖ mid ‖ ote opposite mid is the po nt middle su cohesive mo on to a sca ernal variab imum attain one. e mode I i able per lay
ement midd o interact w nal straight point at the uded becaus on of the eraction of the esive forces. forces have ons of the 3c and 4b) oint betw nd of each ex mid mid e crack fla oint of the c urface. odel describ alar measur le, shown in ned crack o is relevant, yer is intro dle surfaces ith the adva
configurati cutting edg se of the ass cohesive fo blade with th different di truss eleme ). By assum ween blade xplicit time anks, is crack flank bed in [20] re of the cra n figure 4a opening, al , once the oduced betw s. The cable ancing cutti ion of the c ge (the possi sumed shar orce in the (c) he cable eleme irections on ents connec ming that th e and cable e integration s the conta k where the , assuming ack opening for the sim lso interpre fracture ween the element ng blade cable, by ibility of rpness of e correct ents, n the two cting the he blade and the n step, so (13a,13b) act point cable is a linear g ℓ and a mple case ted as a
Figu A soften law is w It is the of the c directio fracture An equ stresses where influenc maximu 5 RES The applicat not trivi during t A thi 5. The polyethy applied constitu
ure 4: (a) soft
ning functio written as fo refore obtai cable. The n normal t e criterion is uivalent plas contribute is the pla ce of triaxia um principa SULTS proposed tions [3,6]. ial because the through-in lamthrough-inate circular ho ylene (HDP by the us uting the hol
(a) tening cohesiv on is a ollows: ℓ ℓ, ined parameter to the open s met; ℓ , i stic strain c to the fractu astic strain ality, and th al stress at th approach h Here we sh of the muc -the-thickne membrane ole to be p PE) cutting ser to the s le does not ve law; (b) dir also introdu 1 2 1 ℓ ℓ 0, 1 ℓ ℓ mid , wh is set e ning faces o s, instead, a criterion, w ure, is adop value at fra he crack dir he node whe has already how an appl ch higher c ess crack pr closes the c produced th teeth has t screw threa contain pap rection of cohe uced so that 1 2 ℓ ℓ 1 ℓ ℓ when ℓ , hile ℓ equal to th of the elem assumed to with a Heav pted: d acture, is rection coin
ere the crite
y proven i lication to a complexity ropagation. carton pack hrough the the diamete ad. In the per, but only
esive forces du , in the line 1 1 2 ℓ ℓ when ℓ , 0. mid he compone ment in the be a materi viside funct s the hydro ncides with erion has be itself prom a multi-laye of the kinem kage for flui
applied cap er of 15 mm remainder y aluminum (b) uring interacti ear softening 0 ent of the s continuum al paramete tion assurin static stress the elemen een met. mising in s ered structur matics of th id aliments p equipped m: it is obta of the pac m and low d
ion with the b
g case, the c mid is th stress tenso m model w er. ng that only s accounting nt edge clos several sing re. The exte he fracture [7] shown i d with high ained with ckage the density poly blade. cohesive (14a-e) he length or in the here the y tensile (15) g for the er to the gle-layer ension is opening in figure h density a torque laminate yethylene
(LDPE); this laminate has a thickness ranging from 70 to 78 m, with the aluminum core 6-9 m thick, while a LDPE layer coat this interior core from both sides.
(a) (b) (c)
Figure 5: (a) package with hole for opening, (b) package with the screw thread applied,
(c) detail of the cutting teeth (upside down view).
In the modelling we assume a circular membrane with radius equal to 10.2 mm and thickness 0.074 mm. The 14% larger diameter accounts for the slackness of the actual membrane due to the manufacturing process. As shown in figure 5c, the cutting tool (referred to as the “blade” from now on) has a radius of 9.4 mm and its edges present a curvature radius of 0.1 mm (see also figure 6b). We consider the blade (figure 6a) as a rigid body whose motion is given: it has a rotation around its vertical axis and a translation along the same axis penetrating the three-layer membrane shown in figure 6b.
(a) (b)
Figure 6: (a) FE model of blade and membrane; (b) detail of the three layers membrane.
The composite laminate in this example is assumed as an equivalent and homogenized
membrane with Young modulus E=176 GPa, Poisson’s ratio =0.3 and density =210-9
Ns2/mm4. A hardening behavior is adopted according to the yielding function
1 exp (27)
where =11 MPa is the initial stress, the saturation parameter is equal to 40 MPa, and the
hardening exponent =6.63. The chosen fracture parameters are 30 N/mm and = 18
MPa. The blades produces several cuts along their circular path as shown in figure 6a; the cut appearance (figure 7a) is in good qualitative agreement with experimental observations of the opening process, as much as the torque vs cap rotation response shown in figure 7b. This comparison shows that, neglecting the initial mismatch between the two curves, due to the
lack of friction imposed on the rotating teeth before contact with the laminate, the agreement is reasonable.
(a) (b)
Figure 7: (a) upside down view of the membrane cut by the blades;
(b) comparison between numerical and experimental torque during the rotation. 6 CONCLUSIONS
A simulation approach for the numerical evaluation of the opening process of a thin-walled laminate has been presented. It makes use of solid-shell elements enriched to correct known locking issues, in a finite strain formulation. To conveniently exploit the advantages of an explicit time integration scheme in structural dynamics, an effective selective mass scaling technique has been implemented, whereby only the highest eigenfrequencies responsible of the smallness of the time step are reduced. To correctly describe the interaction between the blade and the fracturing multi-layered thin-walled membrane, interface directional cohesive elements have been used. We have shown that several layers can be used for the description of the laminate, and fracture initiation and propagation in multi-layered structures is well represented; in the future the possible delamination between layers will be considered.
AKNOWLEDGMENTS
We are grateful for the support given by Tetra Pak Packaging Solutions.
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