Development of a user element in ABAQUS for modelling of cohesive laws in composite structures

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1 Downloaded from orbit.dtu.dk on: Jan 19, 2019 Development of a user element in ABAQUS for modelling of ohesive laws in omposite strutures Feih, Stefanie Publiation date: 2006 Doument Version Publisher's PDF, also known as Version of reord Link bak to DTU Orbit itation (APA): Feih, S. (2006). Development of a user element in ABAQUS for modelling of ohesive laws in omposite strutures. Denmark. Forskningsenter Risoe. Risoe-R, No. 1501(EN) General rights opyright and moral rights for the publiations made aessible in the publi portal are retained by the authors and/or other opyright owners and it is a ondition of aessing publiations that users reognise and abide by the legal requirements assoiated with these rights. Users may download and print one opy of any publiation from the publi portal for the purpose of private study or researh. You may not further distribute the material or use it for any profit-making ativity or ommerial gain You may freely distribute the URL identifying the publiation in the publi portal If you believe that this doument breahes opyright please ontat us providing details, and we will remove aess to the work immediately and investigate your laim.

2 Risø-R-1501(EN) Development of a user element in ABAQUS for modelling of ohesive laws in omposite strutures Stefanie Feih Risø National Laboratory Roskilde Denmark January 2005

3 Author: Stefanie Feih Title: Development of a user element in ABAQUS for modelling of ohesive laws in omposite strutures Department: AFM Risø-R-1501(EN) January 2005 Abstrat (max har.): The influene of different fibre sizings on the strength and frature toughness of omposites was studied by investigating the harateristis of fibre ross-over bridging in DB speimens loaded with pure bending moments. These tests result in bridging laws, whih are obtained by simultaneous measurements of the rak growth resistane and the end opening of the noth. The advantage of this method is that these bridging laws represent material laws independent of the speimen geometry. However, the adaption of the experimentally determined shape to a numerially valid model shape is not straight forward, and most existing publiations onsider theoretial and therefore simpler softening shapes. In this artile, bridging laws were implemented into an interfae element in the UEL user subroutine in the finite element ode ABAQUS. omparison with different experimental data points for rak opening, rak length and rak shape show the sensitivity of these results to the assumed bridging law shape. It is furthermore shown that the numerial preditions an be used to improve the bridging law fit. One shape with one adjustable parameter then fits all experimental data sets. ISSN ISBN ontrat no.: Group's own reg. no.: Sponsorship: over : Pages: 52 Tables: 3 Figures: 26 Referenes: 15 Print: Pitney Bowes Management Servies Denmark A/S, 2005 Risø National Laboratory Information Servie Department P.O.Box 49 DK-4000 Roskilde Denmark Telephone bibl@risoe.dk Fax

4 ontents 1 Introdution 5 2 Mode I bridging law measurement 5 3 Experimental results 6 4 Implementation of the user element Bakground and basi equations Unloading State variables for the analysis Numerial integration 13 5 Numerial model Symmetri half model versus full model: boundary onditions Appliation of pure moment bending by displaement ontrol Bridging law adjustments for numerial modelling 17 6 Numerial results omparison of numerial integration proedures Numerial bridging results with finite stress value Numerial bridging results with power-law inrease Adjustment of bridging law 23 7 Summary 29 A Implementation of the plane interfae element 31 B Quadrati shape funtions and derivatives 33 Integration points and weights 35 D ABAQUS oding for the quadrati line element 37 E ohesive element verifiation 49 3

5 Aknowledgements I would like to thank Bent F. Srensen for readily available input and suggestions regarding ohesive zone models. A speial thanks is going to Lars Pilgaard Mikkelsen for all the help and disussions about finite element analysis and ABAQUS. 4

6 1 Introdution For glass fibre omposites, the interfaial properties are ontrolled by the sizing, whih is applied to the glass fibres during manufature. For the same matrix system, a hange of sizing results in hanges of these properties, thereby influening the mehanial properties suh as strength and frature toughness. The onept of strength is used for haraterising rak initiation in omposite design, while frature toughness determines rak growth and damage development. In mode I rak growth in unidiretional fiber omposites, fibre ross-over bridging ours during raking along the fiber diretion. This failure mode plays an important role during delamination of fibre omposites and splitting raks around holes and nothes. The fibre bridging zone must be modelled as a disrete mehanism on its own; failure is not just ontrolled by the raking at the rak tip. The failure proess an be desribed by a bridging law, whih defines the relationship between the rak opening displaement and the loal bridging trations resulting from the bridging ligaments. ohesive laws were measured experimentally in previous work. This report derives the neessary basis and equations to implement these laws into the ommerial finite element ode ABAQUS with a ohesive user element. Different numerial adjustments of the bridging law are disussed in detail. rak aspets, suh as rak opening shape and the influene of bridging law parameters, are studied based on the numerial results. It is furthermore of interest to identify the experimental measurements whih show the highest sensitivity with respet to the bridging law shape. 2 Mode I bridging law measurement The approah for the measurements of bridging laws is based on the appliation of the path independent J integral [1], and has been used reently to determine the bridging harateristis of unidiretional arbon fibre/ epoxy omposites [2] and glass fibre omposites [3]. A symmetri DB speimen is loaded with pure bending moments M (Figure 1) under pure mode I. This speimen is one of the few pratial speimen geometries, for whih the global J integral (i.e. the integral evaluated around the external boundaries of the speimen) an be determined analytially [1]: M 2 J = 12(1 ν 13 ν 31 ) b 2 H 3 (1) E 11 E 11 is the Young s modulus referring to the material diretions, ν 13 and ν 31 are the major and minor Poisson s ratio, b is the width and H the beam height. M x 2 H+ u 2 * x 1 2H M Figure 1. DB speimen with pure bending moment Now onsider the speimen having a rak with bridging fibres aross the rak faes near the tip. The losure stress σ (x 2 -diretion) an be assumed to depend only on the loal rak 5

7 opening δ, i.e. the rak grows in pure mode I. The bridging law σ = σ(δ) is then taken as idential at eah point along the bridging zone. Sine fibres will fail when loaded suffiiently, we assume the existene of a harateristi rak opening δ 0, beyond whih the losure tration vanishes. Shrinking the path of the J integral to the rak faes and around the rak tip [4] gives J = δ 0 σ(δ)dδ + J tip, (2) where J tip is the J integral evaluated around the rak tip (during raking J tip is equal to the frature energy of the tip, J 0 ). The integral is the energy dissipation in the bridging zone and δ is the end-opening of the bridging zone at the noth root. onnetion is made to the overall R-urves as follows. By definition J R is the value of J during rak growth. Initially, the rak is unbridged. Thus, by Eq. (2), rak growth initiates when J R = J tip = J 0. As the rak grows, J R inreases in aordane with Eq. (2). When the end opening of the bridging zone δ reahes δ 0, the overall R-urve attains its steady state value J ss. The bridging law an be determined by differentiating Eq. (2) [4]. σ(δ ) = J R (3) δ The applied moment and the end opening of the bridging zone u 2 are reorded. Assuming that δ u 2, where u 2 is the noth opening measured at the neutral axis of the DB speimen (see Figure 1), the bridging law an be determined. This approah models the bridging zone as a disrete mehanism on its own. ontrary to rak growth resistane urves (R-urves), the bridging law an be onsidered a material property and does not depend on speimen size [2]. The test above has been modified with two different bending moments to result in mixed mode testing [5]. In this ase, mixed bridging laws an be measured. This has not been undertaken for the urrent material seletion. 3 Experimental results Reently, we have, by the use of a J integral based approah, measured the bridging laws under mode I frature during transverse splitting of unidiretional glass-fiber/epoxy and glass-fiber/ polyester omposites with different interfae harateristis [3]. With inreasing applied moment, rak propagation took plae. Fibre ross-over bridging developed in the zone between the noth and the rak tip. J R is alulated aording to Eq. (1). The speimen width b was 5 mm with a beam height of H=8 mm. Assuming that the unidiretional omposite is transversely isotropi, the following elasti omposite data were applied for Eq. (1) as previously measured: E 11,epoxy = 41.5 GPa, E 33 = 9.2 GPa, E 11,polyester = 42 GPa, E 33,polyester = 10 GPa and ν 13 =0.3 (assumption). The analytial funtion J R (δ ) = J 0 + J ss ( δ δ 0 ) 1 2 was found to fit all experimental data urves of rak growth resistane versus rak opening well, resulting in urve fits as shown in Figure 2. J 0 is the initial value of the experimental urve and equal to the frature energy of the tip during rak growth, while J ss, whih is equal to (J ss J 0 ), is the inrease in rak growth resistane. Sørensen and Jaobsen [2] found that the same funtion fit the data of arbon fibre omposite systems well. The experimental values for the bridging laws are given in Table 1. The starting value J 0 indiates the point of rak growth initiation and an easily be determined during the experiment. The highest value of 345 J/m 2 was observed for the sizing B/ epoxy system. The rak 6 (4)

8 6000 rak growth resistane [J/m 2 ] A / epoxy B / epoxy A / polyester B / polyester rak opening [mm] Figure 2. omparison of rak growth resistane Table 1. Experimental values for the bridging law for different omposite systems omposite system J 0 [J/m 2 ] J ss [J/m 2 ] δ 0 [mm] sizing A/epoxy 321 ± ± ± 0.2 sizing B/epoxy 345 ± ± ± 0.2 sizing A/polyester 150 ± sizing B/polyester 120 ± 30 > 4100 > 5.0 initiation value is signifiantly lower for the sizing B/ polyester system with 120 J/m 2, whih is also related to a signifiantly lower transverse strength of not more than half the strength of the other omposites [3]. In fat, the relation between transverse strength (measured with the transverse bending test [3]) and rak initiation for the different omposite systems is fitted well by a linear relationship as seen in Figure 3. This verifies the assumption that the rak initiation value for the DB test is ontrolled by the strength of the fiber-matrix bond. It an furthermore be seen that both axes show about the same ratio low strength and high strength of nearly a fator 3; however, the transverse bending test results in muh lower standard deviations. The higher standard deviation for the DB frature initiation is most likely explained by speimen-to-speimen differenes of the manufatured noth, whih influenes the rak initiation proess. The end opening value δ 0 at the onset of steady-state raking was determined to be 2 mm for the epoxy systems. For the sizing B/ polyester system, steady-state raking ould not be determined with the present speimens, as the fibres ontinued to bridge the whole length of the rak after the maximum measurable noth opening of 5 mm was obtained. Sine no upper bound was found for J ss, this bridging behaviour was termed infinite toughening. Differentiating equation (4) results in the bridging law σ(δ) = J ss 2δ 0 ( δ δ 0 ) 1 2, for 0 < δ < δ0, (5) where J ss is the inrease in rak growth resistane due to bridging (from zero to steady state bridging), and δ 0 is the rak opening where the bridging stress vanishes. The bridging laws for the different fibre systems are ompared in Figure 4. The bridging law an be onsidered a material property [2, 6] and is in an aessible form for implementation into finite element odes. 7

9 Transverse bending strength [MPa] R 2 = rak initiation J [J/m 2 ] 0 Figure 3. Relationship between rak initiation value J 0 and the transverse omposite strength for the different omposite systems Bridging stress, σ [MPa] A / epoxy B / epoxy A / polyester B / polyester rak opening [mm] Figure 4. omparison of resulting bridging laws 4 Implementation of the user element There are a variety of possible methods for implementing ohesive laws within ommerial finite element programs. The most versatile is the development and programming of ohesive elements [7 10]. These elements are in most ases defined with zero thikness and presribe stresses based on the relative displaement of the nodes of the element. Similar work has also been undertaken with spring elements (fore-opening relation), although in this ase there might be simplifiations required when alulating the equivalent nodal spring fores from the surrounding elements. The proedure is not straight forward when springs are onneted to elements with non-linear shape funtions, suh as 8-noded elements [11]. 8

10 4.1 Bakground and basi equations Figure 5 shows the interfae elements in 2-D and 3-D. The interfae element is made up of two quadrati line elements (a), or two quadrati 2-D plane elements (b). These elements onnet the faes of adjaent elements during the frature proess. The implementation is based on [10]. Quadrati elements are hosen as the antilever beam mostly deforms under bending, whih is best modelled by quadrati solid elements. The nodes of the interfae element need to fit to these elements. The node numbering is hosen aording to ABAQUS onventions of quadrati 2-D and 3D solids. The elements an also be derived in linear form by substituting the quadrati shape funtions with linear ones in the appropriate equations. onsequently, the node numbers and degree of freedoms for the elements will derease η ξ ξ y y x z x Figure 5. (a) Quadrati line interfae element (3 node pairs) and (b) Quadrati plane interfae element (8 node pairs) The two surfaes of the interfae element initially lie together in the unstressed deformation state (zero thikness) and separate as the adjaent elements deform. The relative displaements of the element faes reate normal and shear displaements, whih in turn generate element stresses depending on the onstitutive equations (stress - opening relations) of the material. The onstitutive relationship was derived in the experimental setion, and is independent of the element formulation. The implementation of a general interfae element is explained in the following. Details are given for the speial ase of the quadrati line element for 2-D simulations, while the equivalent formulations for the quadrati plane element for 3-D simulations are provided in Appendix A. The line interfae element has 12 (2x6) degrees of freedom. The vetor (12 1) of the nodal displaements in the global oordinate system is given as: d N = ( d 1 x d1 y d2 x d2 y d6 x d6 y) T (6) The order follows typial ABAQUS onventions, and this is onsidered in the derived formulations below. The opening of the interfae element is defined as the differene in displaements between the top and bottom nodes: u = {u} top {u} bot, (7) thereby leading to the following definition of the interfae opening u N in terms of nodal displaements of paired nodes: u N = Φd N = [ I 6 6 I 6 6 ] d N (8) where I 6 6 denotes a unity matrix with 6 rows and olumns. u N is a 6 1 vetor. From the nodal positions, the rak opening is interpolated to the integration points with the help of standard shape funtions. Let N i (ξ) be the shape funtions for the node pair i (i = 1, 2, 3), where ξ stands for the loal oordinate of the element with 1 ξ 1. The 9

11 relative displaement between the nodes for eah point within the element is then given by: ( ) ux (ξ) u(ξ) = = H(ξ) u N, (9) u y (ξ) where H(ξ) is a 2x6 matrix ontaining the quadrati shape funtions. For the line element, it is of the form ( ) N1 (ξ) 0 N H(ξ) = 2 (ξ) 0 N 3 (ξ) 0. (10) 0 N 1 (ξ) 0 N 2 (ξ) 0 N 3 (ξ) The shape funtions for the line element are given in Appendix B. As a result, we get u(ξ) = H(ξ)Φd N = B(ξ)d N, (11) where B(ξ) is of the dimension 2x12 and u(ξ) of the dimension 2x1; thereby desribing the ontinuous displaement field in both diretions within the element. During large deformations, the element requires a loal oordinate system to ompute loal deformations in normal and tangential diretions. The most ommon hoie is a oordinate system given by the middle points of the two element faes, whih thereby oinides with the nodal positions in the undeformed state. If the oordinates of the initial onfiguration are given by the vetor x N and the deformation state is defined by the vetor d N, the referene surfae oordinates x R N are omputed by linear interpolation between the top and bottom nodes in their deformed state: x N R = 1 2 (I 6 6 I 6 6 )(x N + d N ) (12) The oordinates of any speifi referene plane an be derived analogous to Eq. (9): ( ) x x R (ξ) = R (ξ) y R = H(ξ)x R N (13) (ξ) This loal oordinate vetor, with unit length, is obtained by differentiating the global position vetor with respet to the loal oordinates: t 1 = 1 xr ξ ( ) x R T ξ, yr. (14) ξ The normal vetor (also with unit length) of the loal oordinate element needs to be perpendiular to the vetor t 1 : ( ) 1 y R T t n = xr ξ, xr, (15) ξ ξ and the derivatives are determined as follows: δx R ) (ξ) δξ with h(ξ) = = ( x R,ξ y R,ξ = δ (H(ξ)) x R N = h(ξ)x R N, (16) δξ ( N1,ξ (ξ) 0 N 2,ξ (ξ) 0 N 3,ξ (ξ) 0 0 N 1,ξ (ξ) 0 N 2,ξ (ξ) 0 N 3,ξ (ξ) ). (17) The derivatives of the shape funtions are given in Appendix B. The length of the vetor is given by the standard definition: ( x ) xr ξ = R 2 ( ) y R 2 +. (18) ξ ξ The omponents t 1 and t n represent the diretion osines of the loal oordinate system to the global one, thus defining the 2 2 transformation tensor Θ: 10 Θ = [t 1, t n ], (19)

12 whih relates the loal and global displaements as follows: u lo = Θ T u. (20) In the following, loal matries will be indiated by the subsript lo as above, while the equations otherwise refer to the global values. t lo is the 1 2 vetor defining the bridging stresses in the loal oordinate system and relates to the loal relative displaement via the onstitutive expression for the interfae element: ( ) σ1 t lo = = lo ( u lo ) u lo (21) σ n The onstitutive expression an be expressed either with a linear displaement term for u as shown above, or with a oupled form, where u is inluded with non-linear dependene. The preferred option depends on the form of the onstitutive equation. For our expression as introdued in Eq. (5), a oupled form is preferable: t lo = lo u 1 2 lo (22) lo is now a onstant, and does not depend on the displaement. Note that in omparison with Eq. (5), δ has been replaed with the general numerial nomenlature for the opening, u lo. This onvention will be kept in the following. The element stiffness matrix and the right hand side nodal fore vetor are required for the UEL subroutine in ABAQUS. The element fore vetor is of size Its ontribution to the global fore vetor is defined as f el N = A el B T t da (23) = W B T t dl (24) L el 1 = W B T t detj dξ (25) 1 1 = W B T Θt lo detj dξ, (26) 1 where W is the width of the interfae element and, as in most ases of 2-D modelling, also the width of the finite element model. detj is the Jaobian defined by the transformation of the global oordinates (x, y) to the urrent element oordinate (ξ), and results, for the line element, in the same expression as previously used for alulating the length of the unit vetor in Eq. (18): ( x ) R 2 ( ) y R 2 detj = +. (27) ξ ξ It should be noted that the Jaobian will in most analysis ases not be onstant, but depend on the loal element oordinates. It therefore needs to be derived for eah integration point. The tangent stiffness matrix of aording size (note negative sign onvention for ABAQUS) is defined as f el N K el = d el (28) 11

13 With the derivation from Eq. (23), this results in: K = B T Θ t lo A el d el da (29) = W B T Θ t lo L el d el dl (30) = W B T Θ t lo u lo u L el u lo u d el dl (31) = W B T Θ t lo Θ T B dl (32) L el u lo 1 = W B T ΘD lo Θ T B detj dξ (33) 1 As an be seen above, the stiffness matrix D is defined as: D lo = t lo u lo. (34) It an also be expressed in terms of the onstitutive expression in Eq. (21) with a linear dependene on u: D lo = ( ( u) u) u = ( u) u + ( u) (35) u The loal tration matrix D is then given by ( ) D1 D D lo =. (36) D D n The terms D 1 and D n are derived by finding the derivatives aording to Eq. (34). The omponents D are possible oupling terms. They are normally obtained if the tration laws are derived from an overall elasti potential for the ohesive law. For mode I loading, the relative displaement in u 1 diretion will be zero, and thereby will lead to zero tration stresses. The oupling terms an then be set to zero as they do not influene the results. However, a dummy value for D 1 should be assigned to avoid possible numerial problems due to a singular stiffness matrix. 4.2 Unloading The ohesive relationship as given in Eq. (21) is elasti, and the stresses transferred through the rak obey the same law whether the rak opens or loses. For most damage mehanisms, this is most likely not true, as it neglets the damage introdued at the interfae during partial opening. Loal unloading an our when sudden hanges our in the external loading appliation or internal stress redistribution in neighbouring elements is introdued, for example by sudden failure. This problem is usually irumvented by introduing a maximum damage parameter u, whih stands for the maximum value the opening u obtained during a given inrement of the analysis. If the next value of u is larger, damage ontinues to grow; otherwise elasti unloading is assumed to our (see the elasti unloading towards the origin in the simplified bridging law in Figure 6). This elasti mehanism applies for the ase of fibre bridging. The failure mehanism inludes peeling off of the fibers on either side of the rak, the development of bridging fiber ligaments and failure of these ligaments. During unloading, however, these ligaments simply deform elastially, thereby leading to a elasti unloading towards the origin, whih an be observed during the experiment. For numerial purposes it should be noted that u should only be updated at the end of the inrement; the urrent iterative opening solution is not onsidered. 12

14 σ 0 Bridging stress, σ σ* Elasti unloading σ el =(σ*/ u*) u u* u 0 rak opening u Figure 6. Elasti unloading after damage onset 4.3 State variables for the analysis The number of state variables for eah element depends on the hosen integration proedure. At eah integration point of the linear element, there are two state variables for the relative displaement (x- and y-diretion) and the loal stresses ating in the x- and y-diretion. Furthermore, there is one value desribing the urrent interation state between the two surfaes for the purpose of stability analysis and position traking during the numerial proedure. For a standard integration proedure with 3 integration points, this results in 15 (3x5) state variables for the linear element. As ABAQUS will not terminate the analysis if an insuffiient number of state variables is provided for the hosen integration proedure, this user input error is heked within the UEL subroutine. 4.4 Numerial integration The integration sheme was previously shown to have an influene on the performane of the interfae element. For the Newton-otes sheme, the integration points are loated at the nodes. For linear elements, the appliation of the standard Gauss integration was shown to result in a oupling between the degrees of freedom of different node sets and then in osillations of the tration profile in the presene of large tration gradients, suh as during the initial stiffness inrease prior to damage, over the element [7]. Non-onvergene an our if the element size is too large ompared to the stress uptake based on the tration law: around three elements should be present to resolve the hanging stress state in the interfae [12]. This restrition on maximum element size an also be irumvented by hoosing a larger number of integration points instead of reduing the element size [13]. These approahes are investigated in the report. The 3 point rules (normally hosen as standard in aordane with the quadrati element type) as well as 6 and 12 point rules for Gauss and Newton-otes were tested with the urrent model. The respetive points and weights are given in Appendix. 13

15 5 Numerial model The beam-ontat problem is shown in Figure 7. Owing to symmetry, it is suffiient to onsider F + F = F 2 Initial rak tip H User elements F 1 = M/s s k Free movement at orner point Figure 7. Problem statement and boundary onditions for pure moment loaded DB speimen (symmetri half model) only one beam of the DB speimen, although full models were run to hek the boundary onditions (see Setion 5.1). The thikness H is equal to 8 mm, as disussed in Setion 3. Plane strain onditions were assumed, whih neglet edge effets. The mesh onsisted of eight-noded plane strain solid PE8 elements, whih are suited to desribe bending deformations without hourglass effets. The omposite material is assumed to be transversely isotropi. The material properties were given in the previous setion. Nonlinear, large displaements are onsidered in the analysis to rotate the non-isotropi material properties aordingly. The interfae elements are applied from the rak tip onwards to the end of the beam. The lower element nodes are fixed on the symmetry line. 5.1 Symmetri half model versus full model: boundary onditions For the symmetri half model, additional nodes are dupliated on the symmetry line for the desription of the user elements. During deformation, the top and bottom node sets will separate. The bottom nodes are therefore fixed in the y-diretion as a zero boundary ondition. However, preliminary analyses showed that this ondition does not truly represent the full model (see Figure 8), as the nodes an move freely along the symmetry line in the x-diretion. For the true full model, the bottom nodes will move in the same way as the top nodes, therefore maintaining the same absolute x-value (no shear introdued in the elements). As disussed previously, the oupling terms in the loal onstitutive equation (see Eq. (36)) therefore do not influene the results. For the symmetri half model, the movement of the symmetry nodes in x-diretion therefore needs to be oupled to the movement of the beam nodes via *EQUATION for eah dupliated node. The differene in deformation is visualised in Figure 8. The effet is rather small, but leads to differenes in the results for larger beam defletion. 5.2 Appliation of pure moment bending by displaement ontrol In analyses with possible dereasing stresses due to introdued material damage or dereasing bridging stresses, displaement ontrolled deformation, rather than fore ontrolled, beomes the preferred way of introduing boundary onditions. Furthermore, displaement ontrol is mostly applied in experimental testing and also in the ase of the DB speimen testing: testing proedure and simulation are therefore more losely related. The appliation of a homogeneous bending moment by fore ontrol requires onsideration 14

16 x-diretion movement of symmetry nodes linked to frition parameter (a) Free movement of symmetry nodes Movement of nodes in x-diretion during loading *Equation 2 26, 1, 1, 10000, 1, (b) Linked movement of symmetry nodes x-diretion equation linked for eah node Figure 8. Effekt of linking symmetry nodes of the element type, where varying fores need to be applied to eah node aording to the underlying shape funtions of the element type. This proedure does unfortunately not work for displaements, as for a given translation the rotation of the beam is unknown. The displaement ontrolled proedure introdued in the following was first desribed in previous work of the group [11]. The moment appliation is simplified, thereby resulting in a non-homogeneous displaement field towards the nodes of the applied boundary onditions at the end of the beam. However, pure bending onditions are introdued around the ontat zone by ensuring that the distane k between moment introdution and beginning of the ontat zone (see Figure 7) is large enough. The value is set to k=12 mm in the present model, with s being equal to mm. As a rule of thumb, boundary onditions should be applied more than the beam thikness away from the point of interest to ensure a uniform bending stress in this region. With a beam height of H=8 mm, this requirement is fulfilled. The displaements v 1 and v 2 of two nodes are to be ontrolled suh that the resulting fores in the two nodes, F 1 and F 2, are equal and opposite in magnitude, thereby introduing a pure bending moment (see Figure 9). To aomplish this, v 1 and v 2 are inter-onneted by two soalled dummy nodes A and B, having displaement v A and v B, respetively. Dummy nodes are nodes that are not assoiated with the geometry. Their positions with respet to the beam struture are arbitrary, but their displaements have to be oupled to the strutural deformation. In the following we derive a suitable relationship between v 1, v 2, v A and v B. The displaement of node A is defined as v A = v 1 v 2. (37) 15

17 v 1 θ v 2 s Figure 9. Displaements used for ontrolling the rotation and applying a pure moment to the beam arm of a DB speimen Note that θ = artan(v A /s) v A /s is the rotation of the beam, with s being the urrent differene in x-oordinates between nodes 1 and 2, i.e. the moment arm. The distane between the nodes will hange during the analysis, whih needs to be taken into aount. The displaement in node B is set to v B = v 1 + v 2. (38) These definitions are mostly arbitrary. In a physial sense, they an be desribed as a rotational omponent (displaement ν A ) and a translational omponent (displaement ν B ). We have to make sure that the same energy is applied to the dummy nodes as is applied to the struture itself. Using the priniple of virtual work, the applied inremental elasti energy dw is given by dw = F 1 dv 1 + F 2 dv 2. (39) Similarly, the inremental elasti energy applied to nodes A and B is dw = F A dv A + F B dv B, (40) where F A and F B are the fores in nodes A and B. Finally, the pure bending ondition F 1 + F 2 = 0 (41) must hold true. As the fores in the dummy nodes must (aording to the priniple of virtual work) perform the same inremental work as F 1 and F 2, the following relations are obtained: dw = F A d(v 1 v 2 ) + F B d(v 1 + v 2 ) (42) = (F A + F B )dv 1 + (F B F A )dv 2 (43) In omparison with Eq. (39), this gives the orret relationships between the fores at the nodes. F 1 = F A + F B and F 2 = F B F A (44) 2F A = F 1 F 2 and 2F B = F 1 + F 2 (45) omparing Eq. (41) and Eq. (44) leads to F B = 0, and gives F A = F 1. Thus, the onstraints between v 1, v 2, v A and v B in Eq. (37) and Eq. (38) provide the orret onstraints. The multipoint onstraints for the numerial analysis are implemented into the finite element analysis as follows: 2v 1 v A v B = 0 and 2v 2 + v A v B = 0 (46) In summary, F A is the applied fore to the beam from whih the moment M = F A s = F 1 s an be alulated for eah inrement. J R is omputed from Eq. (1) using the alulated moment. The displaement v A is the loading parameter, whih is inreased during the simulations. 16

18 5.3 Bridging law adjustments for numerial modelling The bridging law adjustment is shown exemplary for the ase of the normal stresses aross the interfae element. With the experimental result of Eq. (5), the stress σ n in normal diretion was defined by the following non-linear relationship σ n = K n u 1 2 n with n = J ss 2 u 0,n for 0 < u n < u 0,n. (47) n is a onstant in this ase as explained in Eq. (22). In the following, we will drop the subsript n for the normal diretion. The relationship above an easily be modified or be expressed in pieewise linear form for different values of u. The use of numerial preditions for a better fit of the ohesive law to the experimental results will be disussed later. Two points need to be addressed during the numerial adjustment: Removal of the stress singularity at u = 0 and Inorporation of the intial frature strength J 0. The bridging law as shown in Eq. (47) has a simple form, but inhibits a singularity in u = 0 : σ (see Eq. 5). Two different methods of numerial adjustment are thought of in the following as shown in Figure 10. Miromehanial onsiderations [6] predit that rak initiation starts at a finite stress value at the interfae, whih is reahed as the deformation starts to take plae. At the point of rak initiation, the opening at the interfae will still be zero (a). This option is unonventional in the numerial sense; and the impliations of using a non-zero starting value for the analysis results have been explored previously [14]. In the ase of a finite stress value σ 0 for u = 0 in the bridging law, the finite element model atually starts with a fore imbalane, as no initial fore equilibrium is given. This is resolved in the first inrement by overlapping the user elements to ahieve zero stresses away from the rak tip. From a numerial point of fore equilibrium at analysis start, a zero stress state is preferable at u = 0 as shown in (b). For small values of u 1 and Deltau 2 these adjustments are only have a small influene on the numerial results. Bridging stress, σ σ 0, A J ss, A σ E,A Bridging stress, σ σ 0, B J ss, B σ E,B u 0 rak opening u u 1, A (a) u 0 rak opening u u 1, B (b) Figure 10. Possibilities of irumventing infinite stress at zero opening For both methods, the most important point is that the energy uptake is equivalent in the experimental and numerial model. Details of the bridging law an then be adjusted in the numerial evaluation to provide the best fit with the experimental data. The differentiation of Eq. (4), whih leads to Eq. (5), results in a loss of the initial starting value J 0 (see Table 1). It is therefore not advisable to simply use the earlier bridging law 17

19 parameters for the numerial simulations. A further adjustment is required. Two proedures based on the initial hoie of singularity treatment above are proposed. The total energy uptake of the experimental bridging law is determined by alulating the area W exp under the urve. σ exp = J ss 2 u 0 ( ) 1 u0 2 u W exp = J ss + J 0 for u [0, u 0 ] (49) For a better physial understanding, J 0 an be onsidered as the elasti frature energy during initial frature (strength ontrolled), while J ss represents the damage frature energy (toughness ontrolled). To ensure the same energy uptake during the simulated frature proess, the initial starting value J 0 needs to be inorporated into the bridging law. For the two bridging law adjustments introdued in Figure 10, different ideas for inluding J 0 are suggested. For method (a) with a finite stress, the initial linear bridging law part an be further adjusted in suh a way that the numerial frature energy at the point u 1 of the bridging proess also inludes the experimental starting value J 0, while for method (b), the easiest method is to split the two omponents J 0 and J ss within the bridging law, and use the initial inrease up to u 2 to introdue J 0. In the following, the neessary adjustments are derived for both ases, and the results are ompared with the experimental data. (48) Finite stress value Figure 11 visualises method (a). Upon reahing the value of u 1, the energy uptake is adjusted Bridging stress, σ σ 0 =(2J J ss u 1 / u 0 )/ u 1 J 0 σ 1 = J ss /(2 u 0 u 1 ) u 1 u 0 0 rak opening δ (a) rak growth resistane [J/m 2 ] Same energy uptake δ 1 Bridging law Adjusted law ( u 1 =0.05 mm) Adjusted law ( u 1 =0.5 mm) rak opening [mm] (b) Figure 11. Numerial bridging law (adjusted). ombined total frature energy. aording to the experimental urve: ( ) 1 u1 2 W exp = J ss + J0 for u [0, u 1 ] (50) u 0 The maximum stress σ 0 is found by requiring that the area W 1 of the linear softening law in the range 0 u u 1 must equal the area W exp of the general bridging law in this range (see Eq. (50)). Area W 1 is given by W 1 = J ss 2 ( u1 u 0 ) σ 0 u 1 (51) 18

20 Setting W exp = W 1, while inluding the starting value J 0 as before, the maximum stress for a given u 1 (as also inluded in Figure 11) is σ 0 = ( ) 1 2J J 2 u1 ss δ 0 u 1 (52) The energy uptake J 0 is now inluded by raising the starting value σ 0. Upon reahing of u 1, the original bridging law is followed. Zero stress value In the seond ase of a zero stress ondition with subsequent inrease of the bridging stresses, a separated bridging law is proposed. A small value of u 2 is assumed, up to whih the energy J 0 is taken up. This is the physially more relevant ase, and is visualised in Figure 12. σ 0 =2J 0 / u 2 = J ss /(2 u 0 u 2 ) σ( u)= u/ u 2 σ 0 σ 0 =J 0 / u 2 ((α+1)/α)= J ss /(2 u 0 u 2 ) σ( u)=(1-(( u 2 - u)/ u 2 ) α ) σ 0 Bridging stress, σ J 0 u2 σ( u)= J ss,adj /(2 u 0 u ) σ Bridging stress, J ss J 0 u2 σ( u)= J ss, adj /(2 u 0 u ) J ss u 0 rak opening u rak opening u u 0 (a) Initial linear inrease (b) Initial power-law inrease Figure 12. Numerial bridging law (adjusted). Separated frature energy. The value of u 2 is determined from the initial frature energy of J 0 and the maximum stress value, σ 0. u 2 an beome quite small based on the value of J 0. This leads to onvergene problems with the linear inrease shown in Figure 12(a), as a steep tangent hanges to a negative tangent at u = u 2. Instead of the ommon linear inrease up to u 2, a power-law approah (see Figure 12(b)) an be applied: [ ] α ) u2 u σ( u) = σ 0 (1 (53) u 2 u 2 an then be determined for a given value of J 0 by the integral: u2 0 σ( u)d u = J 0 (54) u 2 = J 0 σ 0 ( ) α + 1 α The power-law fator α determines the initial inrease in stress with opening. The inrease with α = 1 is idential to the linear inrease. A fator of α = 100 leads to the same result for u 2 as a onstant finite stress value up to u 2 with a non-equilibrium starting point as shown in Table 2. The proposed power-law funtion is therefore more versatile. However, the power-law approah is also a numerially more robust approah as the sign hange in the tangent stiffness (55) 19

21 Table 2. omparison of values for u 2 for a maximum stress of σ 0 = 3.5 MPa and J 0 = 300 J/m 2 α=1 α=5 α=10 α=100 linear onst σ 0 u 2 [mm] at u = u 2 is avoided. The tangent is lose to zero before beoming negative. This leads to largely improved numerial behaviour in terms of onvergene. As the energy uptake for the damage part needs to equal the original J ss for whole bridging law between 0 and u 0, the value of J ss, adj (see Figure 12) in the bridging law needs to be adjusted aordingly: J ss, adj = 1 J ss u 2 u 0 (56) The problem with this proedure is that σ 0 in turn again depends on the adjusted value for J ss, adj : σ 0 = J ss, adj 2 u 2 u 0 (57) As a onsequene, σ 0 and therefore u 2 will hange as well. An iterative proedure needs to be applied to determine the values for J ss, adj and u 2. A starting value for u 2 in Eq. (56) an be determined by integrating the bridging law within the limits of u 2 and u 0 and setting the integral value equal to J ss (see Figure 12 (b)): ( ) 2 α + 1. (58) u 2 = 4J 0 2 u 0 Jss 2 α We therefore determine first u 2 with Eq. (58). This is followed by alulating J ss, adj with Eq. (56) and σ 0 is alulated with Eq. (57). In the last step, u 2 is determined with Eq. (55). The latter three steps are repeated until onvergene is ahieved. 6 Numerial results In the following setions, results are introdued for the bridging law shape with finite stress value (method (a)) and the bridging law shape with split values of J 0 and J ss with a powerlaw inrease (method (b)) as introdued earlier. Based on these results, the bridging law shape was then modified to give a better fit with the experimental data. To justify this proedure, it is important to keep in mind that the bridging law was derived in the first plae by analytial urve fitting of the rak growth resistane (see Setion 2). Any deviation from the experimental data to this urve fitting will of ourse influene the math of the preditions. Analytially, it is only possible to fit the rak growth resistane. In this setion, we also ompare the numerial fit with other experimental observations, suh as the rak growth resistane as a funtion of rak growth development and the shape of the rak opening. Based on these findings, it is shown that the rak growth resistane as a funtion of rak growth development is muh more sensitive to hanges in the bridging law shape. Numerial adjustment of the bridging law parameters on this urve leads to a very good agreement of the numerial preditions with all measured experimental data. 6.1 omparison of numerial integration proedures Non-onvergene problems an our in the following simulations if the element size is too large ompared to the stress uptake based on the tration law. This is most severe for the method of the finite stress value, while all approahes with the power-law show a better numerial 20

22 performane. This restrition on element size an be improved by hoosing a larger number of integration points instead of reduing the element size. Different mesh refinements were used to determine the numerial response during the simulations. For the first refinement, whih was shown in Figure 8, the element length for the ohesive elements is given by 0.36 mm. This is in relation to values of u 1 = 0.05 mm and u mm. The mesh density is not suffiient to ensure onvergene for 3 Gauss integration points and Newton-otes points for all methods introdued in the following: further mesh refinement for those integration proedures is required down to l el = mm is required. At this level, all integration proedures onverge. However, the use of 6 integration points for both the Gauss and Newton-otes integration result in onvergene of the simulation. Running the simulation with a lower mesh density is of ourse benefiial in terms of omputational effort. For 12 integration points, Gauss integration again shows good performane, while the 12 point Newton-otes integration does not onverge after an initial start of the analysis. Some osillatory effets in the numerial preditions of the steady-state level are observed for a large number of integration points and low mesh density. 6.2 Numerial bridging results with finite stress value alulations were undertaken by method (a) with a finite stress value. The value of J 0 was inluded as desribed earlier. Two experimental tests were simulated: sizing A / epoxy and sizing A / polyester. As speifi experiments were modelled, the data was not fitted to the average of all tests as shown in Table 1, but to the speifi test data of some of the speimens (one for sizing A / epoxy, two for sizing A / polyester). For these speimens, we also reorded the rak length as a funtion of rak growth resistane. The following data were used for the bridging law (for referene for the numerial nomenlature see Figure 11): Sizing A / epoxy: J ss =3300 J/m 2 J 0 =300 J/m 2 u 0 = 2 mm u 1 = 0.05 mm σ 0 = MPa (σ 1 = 5.22 MPa) Sizing A / polyester: J ss =4000 J/m 2 J 0 =130 J/m 2 u 0 = 5.5 mm u 1 = 0.05 mm σ 0 = MPa (σ 1 = 3.81 MPa) The value for u 1 has to be hosen as small as possible if we want to imitate the original bridging law shape. However, there are numerial limitations, as with dereasing u 1 the value of σ 0 inreases rapidly. The urrent value of u 1 = 0.05 mm was found to lead to onverging numerial studies with suitable levels of mesh refinement. The fit with the rak growth resistane as shown in Figure 13 is satisfatory apart from slight differenes in the initial inrease of the rak growth resistane for small openings. For both materials, the model predits a steeper inrease than experimentally observed. Figure 14 shows the omparison of the numerial preditions and the measured rak length for different values of rak growth resistane. The urrent numerial rak length is determined from a minimum positive node opening at the rak tip. An small threshold value of 1e-4 is used to determine the rak tip position. The results are quite insensitive to threshold value variations between 1e-4 and 1e-8. It an be learly seen that this omparison is muh more sensitive to the bridging law shape, and the figure shows the defiieny of the urrent bridging law shape learly. Numerially, for 21

23 rak growth resistane [J/m 2 ] J ss = 3300 J/m 2 J 0 = 300 J/m 2 u 0 = 2 mm u 1 = 0.05 mm J ss = 4000 J/m 2 J 0 = 130 J/m 2 u 0 = 5.5 mm u 1 = 0.05 mm sizing A/epoxy 1000 Data sizing A/epoxy sizing A/polyester Data sizing A/polyester rak opening [mm] Figure 13. omparison of rak growth resistane for method (a) with finite stress value rak growth resistane [J/m 2 ] sizing A / epoxy sizing A / polyester J ss = 3300 J/m 2 J 0 = 300 J/m 2 u 0 = 2 mm u 1 = 0.05 mm J ss = 4000 J/m 2 J 0 = 130 J/m 2 u 0 = 5.5 mm u 1 = 0.05 mm rak extension, a [mm] Figure 14. omparison of rak growth development for method (a) with finite stress value a given rak length a higher rak growth resistane is predited. The seond onern is the initial predition of rak growth. Naturally, due to the adjustment of the bridging law with respet to the inorporation of J 0 into the initial linear part, rak growth starts at zero rak growth resistane, and reahes the value after the initial rak opening of u 1 = 0.05 mm. This orresponds to approximate rak lengths of 5 mm for epoxy, and 3 mm for the polyester as indiated by the knee in Figure Numerial bridging results with power-law inrease We proposed splitting the values of J 0 and J ss with this method as shown in Figure 12. Using an iterative proedure with Eq. (56), (57) and (55) as explained previously, we get the following input values for the bridging law: Sizing A / epoxy: u 0 = 2 mm α =

24 J ss = 3300 J/m 2 J 0 =300 J/m 2 u 2 = mm, σ 0 = 6.29 MPa and J ss, adj =3906 J/m 2 Sizing A / polyester: u 0 = 5.5 mm α = 100 J ss = 4000 J/m 2 J 0 =130 J/m 2 u 2 = mm, σ 0 = 6.29 MPa and J ss, adj =4262 J/m 2 The determined values for u 2 are small in omparison with u 0 ( u0 u 2 >20). It was therefore deided to keep u 0 onstant. Figure 15 shows the omparison between numerial and experimental data for the rak growth resistane as a funtion of rak opening. It an be seen that the numerial preditions for small rak openings fit better with the splitted damage energies than it did with method (a). rak growth resistane [J/m 2 ] J ss = 3300 J/m 2 J 0 = 300 J/m 2 u 0 = 2 mm J ss = 4000 J/m 2 J 0 = 130 J/m 2 u 0 = 5.5 mm sizing A/epoxy 1000 Data sizing A/epoxy sizing A/polyester Data sizing A/polyester rak opening [mm] Figure 15. omparison of rak growth resistane for method (b) with power-law inrease However, although the fit in Figure 15 seems nearly perfet with the proposed bridging law, the rak length as a funtion of the rak opening still shows deviations between the numerial and experimental results (see Figure 16). The urrent numerial rak length is determined from a minimum positive node opening at the rak tip, whih is now defined as the value of u 2, and thereby orresponds to rak development after reahing the rak initiation energy of J 0. In the opinion of the author, this assumption has a physially more valid bakground than a randomly hosen small threshold value: one an onsider damage models where frature is aused by initial mirovoids forming and growing together [15]. Upon reahing the value of damage initiation J 0, these mirovoids will form a starting rak. 6.4 Adjustment of bridging law The above examples showed that the bridging law shape an be further improved to provide a better fit with the experimental data; espeially with respet to the rak length as a funtion of rak growth resistane. Previous trials with the above models were taken as the basis to improve the fit. Based on the above results, the following onlusions are drawn: Splitting of the bridging law into an initial frature (J 0 ) and damage ( J ss ) ontrolled part is of advantage based on the physial interpretation of the ohesive law parameters. For 23

25 rak growth resistane [J/m 2 ] sizing A / epoxy sizing A / polyester J ss = 3906 J/m 2 J 0 = 300 J/m 2 u 0 = 2 mm u 1 = 0.05 mm J ss = 4262 J/m 2 J 0 = 130 J/m 2 u 0 = 5.5 mm u 2 = 0.02 mm rak extension, a [mm] Figure 16. omparison of rak growth development for method (b) with power-law inrease example, for a failure mode suh as delamination between arbon / epoxy prepreg plies, the initial frature energy surpasses the damage energy; while the bridging phenomenon of unidiretional glass fiber omposites demonstrates the opposite material behaviour. Bridging law values should be derived from the experimental urves instead of being subjet to a wide-ranging trial-and-error exerise. It is therefore desirable to estimate the value of u 2 from the experimental data diretly. This would be even more important if u 2 is not small ompared to u 0, i.e. both J 0 and J ss have omparable order of magnitude. For method (b) splitting of initial frature (J 0 ) and damage ( J ss ) frature energy improves the fit with the experimental data. However, the initial drop in the bridging law (for values larger than u 2 ) is still too severe; thereby resulting in an overpredition of the rak growth resistane for a ertain rak length. This hanges to the bridging law shape are visualised in Figure 17. The shape is basially a ombination of the previously disussed shapes for method (a) and (b). For K fa =1 and u 1 =0 we reset the bridging law to the shape of method (b). The derivation of these values is explained below. Determination of u 2 Previously, the value for u 2 was derived with a nonlinear solving proedure. However, as the value atually has a physial meaning as explained in the last setion, the experimental urves an be used to determine suitable values. This is shown in Figure 18. The figure shows that prior to rak initiation, the experimental urves atually exhibit a small displaement of the order of µm. Upon reahing J 0, these mirovoids will start to form the starting rak. The orresponding rak growth resistane is the value of J 0 as used previously (J 0,epoxy =300 J/m 2, J 0,poly =130 J/m 2 ). The orresponding displaement value an be used for u 2. Based on Eq. (55), we an then alulate a value for the required peak stress and thereby introdue a fator value K fa in omparison to the original peak stress: K fa = J 0 u 2 σ 0 ( ) α + 1 α (59) Determination of u 1 To improve the fit with the experimental data further, the dereasing bridging law slope after 24

26 σ 0 =J 0 /( u 2 K fa )((α+1)/α) σ(δ)=(1-(( u 2 - u)/ u 2 ) α ) K fa σ 0 σ 2 =1.5 J ss u 1 / u 0 / u 1 Bridging stress, σ J 0 u2 σ 1 = J ss /(2 u 0 u 1 ) J ss u 1 u 0 rak opening u Figure 17. Adjusted bridging law 500 rak growth resistane [J/m 2 ] u 2 =10µm J 0 =300 J/m u 2 =8µm 0 J 0 =130 J/m rak opening [mm] Figure 18. Determine u 2 experimentally reahing u 2 needs to be redued. We therefore take over the previously introdued linear derease between u 2 and u 1 in this model. This hoie of an initially linear derease is also based on previous numerial parameter studies of bridging law shapes [4, 11]. Bridging laws with linear derease then show an s- shaped trend in the rak length development. Judging from the experimental data, where we overpredit more in the initial rak development stages, this type of model would improve the fit with the experimental data. The value of u 1 now beomes the only adjustable value in the bridging law to be fitted to the experimental data points. Determination of σ 0 σ 0 an be alulated from the requirement of total energy update equal to J ss as previously with model (a), without inluding J 0 as this part of the frature energy is taken are of during 25

27 the initial inrease up to u 2. σ 0 = ( ) J 2 u 1 ss δ 0 u 1 (60) Bridging law parameters J 0, J ss, u 2 and u 0 an all be determined diretly from the experimental data. The values are summarised in Table 3. Again, the value of u 2 is small in omparison with u 0, so there is no differene in the numerial predition as to whether the length of the bridging law is given by u 0 or u 0 + u 2. Table 3. Numerial values for the bridging law for different omposite systems omposite system J 0 J ss K fa u 0 u 2 u 1 [J/m 2 ] [J/m 2 ] [mm] [mm] [mm] sizing A/epoxy sizing A/polyester Figure 19 shows the resulting fit for the rak growth resistane versus rak opening where the value of u 1 has been adjusted to 1.0 and 2.5 mm for epoxy and polyester, respetively. In both ases, this is about half the value of the total rak opening, u 0. rak growth resistane [J/m 2 ] J ss = 3300 J/m 2 J 0 = 300 J/m 2 u 0 = 2 mm u 1 = 1 mm J ss = 4000 J/m 2 J 0 = 130 J/m 2 u 0 = 5.5 mm u 1 = 2.5 mm sizing A/epoxy 1000 Data sizing A/epoxy sizing A/polyester Data sizing A/polyester rak opening [mm] Figure 19. omparison of rak growth resistane Figure 20 now shows the exellent fit between the numerial and experimental data for the rak growth resistane versus rak length. The s-shaped rak length urve is now predited extremely well. For a further hek of the proposed rak bridging shape, the predited rak profiles are studied together with the experimental preditions. This was undertaken for the polyester speimens as seen in Figure 21. The figure shows the rak opening profile for a rak length of 50 mm. For auray of the experimental rak opening measurements, an opening of at least 0.2 mm must be given. It an be seen that the bridging law also in this ase leads to a very good agreement between the two data sets. Additionally, the unbridged rak profile is shown in the figure. The bridging stresses aross the rak faes lead - as expeted - to a derease in the opening of the rak. 26

28 rak growth resistane [J/m 2 ] sizing A / epoxy sizing A / polyester J ss = 3300 J/m 2 J 0 = 300 J/m 2 u 0 = 2 mm u 1 = 1 mm J ss = 4000 J/m 2 J 0 = 130 J/m 2 u 0 = 5.5 mm u 1 = 2.5 mm rak extension, a [mm] Figure 20. omparison of rak growth development rak opening [mm] sizing A / polyester unbridged profile J R =3100 J/m rak position [mm] Figure 21. omparison between numerial and experimental data for the rak opening profile for sizing A / polyester The maximum opening at the rak noth is - for a length of 50 mm - redued to 25% from 2.0 to 1.5 mm. This is quite a onsiderable effet of the bridging stresses. Figure 22 shows the results from above together with the numerial preditions for sizing A/epoxy at the same rak growth resistane of J R =3100 J/m 2. It an be seen that the rak opening and rak length for the same rak growth resistane are onsiderably smaller for sizing A/epoxy. Inluding the bridging stresses also hanges the rak opening shape in the viinity of the rak tip - however, this hange is hard to determine orretly experimentally due to the small rak opening values. Figure 23 now shows the numerial preditions for the rak opening profiles for both the sizing A/epoxy and sizing A/polyester system in omparison for the same rak opening. The urves are plotted for two different rak opening values of 0.3 and 1.2 mm. For these rak openings, sizing A/polyester gives the larger rak length together with a smaller rak growth resistane. 27

29 rak opening [mm] sizing A / epoxy sizing A / polyester rak position [mm] Figure 22. omparison rak profiles for different bridging laws with J R =3100 J/m 2 rak opening [mm] sizing A / epoxy sizing A / polyester J R = 2780 J/m 2 J R = 1030 J/m 2 J R = 3640 J/m 2 J R = 2010 J/m rak position [mm] Figure 23. omparison of rak profiles for different bridging laws Unloading onsiderations A possible proedure for unloading (i.e. rak losure during strutural loading) was desribed earlier in Setion 4.1 (see Figure 6). Although the urrent problem situation does not require an unloading proedure due to ontinually inreasing rak opening, a theoretial onsideration is given here. The splitting of the two frature parts, J 0 and J ss, an then also be used to hange the history for elasti unloading: if we onsider frature initiation ompleted one the value of u 2 has been reahed (void formation out of mirovoids), elasti unloading to the origin should only our for openings larger than u 2. For smaller values, the path an simply be reversed. This has not been investigated in detail as the urrent problem does not require unloading. 28

30 7 Summary This report desribes the implementation of a ohesive user element into the ommerial FE ode ABAQUS. This element type proves to be a versatile tool in prediting opening, rak length and rak profiles as a funtion of the rak growth resistane. In omparison with the earlier implemented UINTER ontat routine, the ohesive element has the distint advantage that a higher number of integration points an be used to improve the numerial performane for a given mesh density. This redues the omputational time for the analysis greatly. The experimentally measured bridging law was optimised with the help of the numerial preditions. Experimental data suh as rak length and rak profiles were found to give very good agreement with the adjusted law. The parameters for the new law an mostly be determined from the experiments; only one parameter remains to be fitted. 29

31 Referenes [1] J.R. Rie. A path independent integral and the approximate analysis of strain onentration by nothes and raks. Journal of Applied Mehanis, 35: , [2] B.F. Sørensen and T.K. Jaobsen. Large sale bridging in omposites: R-urves and bridging laws. omposites: Part A, 29A: , [3] S. Feih, J. Wei, P.K Kingshott, and B.F. Sørensen. The influene of fibre sizing on strength and frature toughness of glass fibre reinfored omposites. omposites: Part A, 36(2): , [4] Z. Suo, G. Bao, and B. Fan. Delamination R-urve phenomena due to damage. Journal of Mehanis and Physis in Solids, 1(40):1 16, [5] B.F. Sørensen, K. Jørgensen, T.K. Jaobsen, and R.. Østergaard. A general mixed mode frature mehanis test speimen: The DB-speimen loaded with uneven bending moments. Tehnial Report Risø-R-1394(EN), Risø National Laboratory, Roskilde, Denmark, Marh [6] S.M. Spearing and A.G. Evans. The role of fibre bridging in the delamination resistane of fiber-reinfored omposites. Ata Metallia Materials, 40(9): , [7] J..J. Shellekens and R. De Borst. On the numerial integration of interfae elements. International Journal for Numerial Methods in Engineering, 36:43 66, [8].G. Davila, P.P. amanho, and M.F. de Moura. Mixed-mode deohesion elements for analyses of progressive delamination. In 42nd AIAA/ASME/ASE/AHS/AS Strutures, Strutural Dynamis and Materials onferene, number AIAA , [9] J. hen, M. risfield, A.J. Kinloh, E.P. Busso, F.L. Matthews, and Y. Qiu. Prediting progressive delamination of omposite material speimens via interfae elements. Mehanis of omposite materials and strutures, 6: , [10] J. Segurada and J. LLora. A new three-dimensional interfae element to simulate frature in omposites. International Journal of Solids and Strutures, 41: , [11] T.K. Jaobsen and B.F. Sørensen. Mode I intra-laminar rak growth in omposites - modelling of R-urves measured from bridging laws. omposites: Part A, 32:1 11, [12] Y. Mi, M.A. risfield, and G.A.O. Davies. Progressive delamination using interfae elements. Journal of omposite Materials, 32: , [13] G. Alfano and M.A. hrisfied. Finite element interfae models for the delamination analysis of laminated omposites: mehanial and omputational issues. International Journal for numerial methods in engineering, 50: , [14] S. Feih. Modelling ohesive laws in finite element simulations via an adapted ontat proedure in ABAQUS. Tehnial Report Risø-R-1463(EN), Risø National Laboratory, Roskilde, Denmark, July [15] A. orne, I. Shneider, and K.H. Shwalbe. On the pratial appliation of the ohesive model. Engineering frature mehanis, 70: ,

32 A Implementation of the plane interfae element A.1 Derivation of fore vetor and stiffness matrix The derivation of the fore vetor and stiffness matrix for the plane interfae element are analogeous to the linear element presented earlier. The plane interfae element has 48 (3x16) degrees of freedom. The nodal displaements in the global oordinate system are given as: d N = ( d 1 x d1 y d1 z d2 x d2 y d2 z... ) T d16 x d16 y d16 z (A.1) u N = Φ d N = ( I I ) d N, (A.2) where I denotes a unity matrix with 24 rows and olumns. Let N i (ξ, η) be the shape funtions for the node pair i (i = 1,..., 8), where ξ and η stand for the loal oordinates of the element with 1 ξ 1 and 1 η 1. The relative displaement between the nodes for eah point within the element is then given by: u(ξ, η) = u x (ξ, η) u y (ξ, η) u z (ξ, η) = H(ξ, η) u N, where H(ξ, η) is a 3x24 matrix ontaining the individual shape funtions. It has the form (A.3) H(ξ, η) = (N 1 (ξ, η) I 3x3 N 2 (ξ, η) I 3x3 N 8 (ξ, η) I 3x3 ). (A.4) for plane interfae elements. As a result, we get u(ξ, η) = HΦd N = Bd N, where B is of the dimension 3x48 and u of the dimension 3x1. (A.5) The oordinates of any speifi referene position an be derived aording to Eq. (A.3): x R (ξ, η) = H(ξ, η)x R N (A.6) The tangential plane is spanned by two vetors, v ξ and v η. They are, at a given point, obtained by differentiating the global position vetor with respet to the loal oordinates. Although v ξ and v η are generally not orthogonal to eah other, their vetor produt defines the surfae normal. Therefore, the loal normal vetor is obtained by: t n = xr ξ 1 xr η ( x R ξ xr η The tangential oordinates are defined as: t 1 = 1 xr xr ξ ξ ) (A.7) (A.8) t 2 = t n t 1 (A.9) The omponents t 1, t 2 and t n represent the diretion osines of the loal oordinate system to the global one, thus defining the 3 3 transformation tensor Θ: Θ = (t 1, t 2, t n ). (A.10) The loal displaements are then obtained by u lo = Θ T u. (A.11) 31

33 The element stiffness matrix and the right hand side nodal fore vetor are required for the UEL subroutine in ABAQUS. The nodal fore vetor is defined as fn el = B T Θ T t lo det J dξdη, (A.12) where t lo is the 3x1 vetor defining the bridging stresses. The bridging stresses are onneted to the opening displaements by the ohesive law as derived previously. It should be notied that - for numerial purposes - the relationship an be expressed in various forms (analytial, pieewise linear, et). det J is the Jaobian defined by the transformation of the urrent element oordinates (ξ, η) to the global oordinates (x, y, z), and defined as det J = (det J1) 2 + (det J2) 2 + (det J3) 3 with (A.13) det J1 = x R ξ y R η x R η y R ξ (A.14) det J2 = x R ξ z R η x R η z R ξ (A.15) det J3 = y R ξ z R η y R η z R ξ (A.16) Note that depending on the orientation of the loal to the global oordinate system, the omponents det J1, det J2 and det J3 an be negative. The tangent stiffness matrix (note sign onvention for ABAQUS) is defined as K el el fn = d el = B T Θ T D lo ΘB ds el K = el B T Θ T D lo ΘB dξdη (A.17) (A.18) A.2 State variables for the analysis The number of state variables for eah element depends on the hosen integration proedure. At eah integration point of the plane element, there are three state variables for the relative displaement (x-, y- and z-diretion) and the loal stresses ating in the x-, y- and z-diretion. Furthermore, there is one value desribing the urrent interation state between the two surfaes for the purpose of stability analysis and position traking during the numerial proedure. For a standard integration proedure with 9 integration points, this results in 63 (9x7) state variables for the plane element. As ABAQUS will not terminate the analysis if an insuffiient number of state variables is provided for the hosen integration proedure, this user input error is heked within the UEL subroutine. 32

34 B Quadrati shape funtions and derivatives B.1 Line element The three quadrati shape funtions for line element are as follows: N 1 (ξ) = 1 2 ( ξ + ξ2 ) (B.1) N 2 (ξ) = 1 2 (ξ + ξ2 ) (B.2) N 3 (ξ) = 1 ξ 2 (B.3) Derivative of shape funtions N 1,ξ (ξ) = ξ (B.4) N 2,ξ (ξ) = ξ (B.5) N 3,ξ (ξ) = 2ξ (B.6) B.2 Plane element The quadrati shape funtions for the plane element are given below: N 1 (ξ, η) = 1 4 (1 ξ)(1 η) 1 2 (N 5 + N 8 ) (B.7) N 2 (ξ, η) = 1 4 (1 + ξ)(1 η) 1 2 (N 5 + N 6 ) (B.8) N 3 (ξ, η) = 1 4 (1 + ξ)(1 + η) 1 2 (N 6 + N 7 ) (B.9) N 4 (ξ, η) = 1 4 (1 ξ)(1 + η) 1 2 (N 7 + N 8 ) (B.10) N 5 (ξ, η) = 1 2 (1 ξ2 )(1 η) (B.11) N 6 (ξ, η) = 1 2 (1 + ξ)(1 η2 ) (B.12) N 7 (ξ, η) = 1 2 (1 ξ2 )(1 + η) (B.13) N 8 (ξ, η) = 1 2 (1 ξ)(1 η2 ) (B.14) 33

35 Derivative of shape funtions N 1,ξ = 1 4 (1 η) 1 2 (N 5,ξ + N 8,ξ ) (B.15) N 2,ξ = 1 4 (1 η) 1 2 (N 5,ξ + N 6,ξ ) (B.16) N 3,ξ = 1 4 (1 + η) 1 2 (N 6,ξ + N 7,ξ ) (B.17) N 4,ξ = 1 4 (1 + η) 1 2 (N 7,ξ + N 8,ξ ) (B.18) N 5,ξ = ξ(1 η) (B.19) N 6,ξ = 1 2 (1 η2 ) (B.20) N 7,ξ = ξ(1 + η) (B.21) N 8,ξ = 1 2 (1 η2 ) (B.22) N 1,η = 1 4 (1 ξ) 1 2 (N 5,η + N 8,η ) (B.23) N 2,η = 1 4 (1 + ξ) 1 2 (N 5,η + N 6,η ) (B.24) N 3,η = 1 4 (1 + ξ) 1 2 (N 6,η + N 7,η ) (B.25) N 4,η = 1 4 (1 ξ) 1 2 (N 7,η + N 8,η ) (B.26) N 5,η = 1 2 (1 ξ2 ) (B.27) N 6,η = η(1 + ξ) (B.28) N 7,η = 1 2 (1 ξ2 ) (B.29) N 8,η = η(1 ξ) (B.30) 34

36 Integration points and weights.1 Gauss integration point positions and weights 3 points p 1 = 0.6 w 1 = p 2 = 0.0 w 2 = p 3 = 0.6 w 3 = points p 1 = w 1 = p 2 = w 2 = p 3 = w 3 = p 4 = w 4 = p 5 = w 5 = p 6 = w 6 = points p 1 = w 1 = p 2 = w 2 = p 3 = w 3 = p 4 = w 4 = p 5 = w 5 = p 6 = w 6 = p 7 = w 7 = p 8 = w 8 = p 9 = w 9 = p 10 = w 10 = p 11 = w 11 = p 12 = w 12 =

37 .2 Newton-otes integration point positions and weights 3 points p 1 = 1.0 w 1 = p 2 = 0.0 w 2 = p 3 = 1.0 w 3 = points p 1 = 1.0 w 1 = 19/144 p 2 = 3/5 w 2 = 75/144 p 3 = 1/5 w 3 = 50/144 p 4 = 1/5 w 4 = 50/144 p 5 = 3/5 w 5 = 75/144 p 6 = 1.0 w 6 = 19/ points p 1 = 1.0 w 1 = / p 2 = 9/11 w 2 = / p 3 = 7/11 w 3 = / p 4 = 5/11 w 4 = / p 5 = 3/11 w 5 = / p 6 = 1/11 w 6 = / p 7 = 1/11 w 7 = / p 8 = 3/11 w 8 = / p 9 = 5/11 w 9 = / p 10 = 7/11 w 10 = / p 11 = 9/11 w 11 = / p 12 = 1.0 w 12 = /

38 D ABAQUS oding for the quadrati line element The following ode will be ompiled and linked when alling the ABAQUS job. For unix systems, the ommand prompt will be as follows: abaqus job=jobname user=interfae This syntax is of ourse dependent on the platform you are using. For further information see the ABAQUS manuals. The following ode for the user subroutine UEL is written in FOR- TRAN 77 and saved as interfae.f. SUBROUTINE UEL(RHS,AMATRX,SVARS,ENERGY,NDOFEL,NRHS,NSVARS, 1 PROPS,NPROPS,OORDS,MRD,NNODE,U,DU,V,A,JTYPE,TIME,DTIME, 2 KSTEP,KIN,JELEM,PARAMS,NDLOAD,JDLTYP,ADLMAG,PREDEF, 3 NPREDF,LFLAGS,MLVARX,DDLMAG,MDLOAD,PNEWDT,JPROPS,NJPROP, 4 PERIOD) INLUDE ABA_PARAM.IN PARAMETER (ZERO = 0.D0, HALF=0.5D0, ONE= 1.0D0, TWO=2.0d0, 1 THREE= 3.0d0, TOL=-1E-5) DIMENSION RHS(MLVARX,*),AMATRX(NDOFEL,NDOFEL), 1 SVARS(NSVARS),ENERGY(8),PROPS(*),OORDS(MRD,NNODE), 2 U(NDOFEL),DU(MLVARX,*),V(NDOFEL),A(NDOFEL),TIME(2), 3 PARAMS(3),JDLTYP(MDLOAD,*),ADLMAG(MDLOAD,*), 4 DDLMAG(MDLOAD,*),PREDEF(2,NPREDF,NNODE),LFLAGS(*), 5 JPROPS(*) GENERAL ELEMENT VALUES DIMENSION STRESS(MRD) DIMENSION DDSDDR(MRD,MRD) GAUSS INTEGRATION VARIABLES (3 INTEG POINT) DIMENSION GAUSS3(3), WEIGHT3(3), OTNEW(3), WEIGHT(3) DIMENSION GAUSS6(6), WEIGHT6(6), OTNEW6(6), W6(6) DIMENSION GAUSS12(12), WEIGHT12(12), OTNEW12(12), W12(12) ARRAYS FOR QUADRATI LINE ELEMENT DIMENSION DNDXI(3), DELTA_U(6), DU_ONT(MRD), DU_LO(MRD) DIMENSION H(MRD,6), _OOR(MRD,NNODE), PSI(6,NDOFEL) DIMENSION B(MRD, NDOFEL), BT(NDOFEL, MRD) DIMENSION A1(NDOFEL, MRD), A2(NDOFEL, NDOFEL) DIMENSION AV_OOR(MRD, 3), V_XI(MRD), V_N(MRD) DIMENSION THETA(MRD, MRD), STR_GLOB(MRD) DIMENSION D_GLOB(MRD, MRD), DD1(MRD, MRD) data iuel/0/ save iuel QUADRATI LINE ELEMENT SVARS - In 1, ontains the lopenlose identifier - In 3-4, ontains the tration stiffness - In 5-6, ontains the tration opening INITIALISATION: IMPORTANT!! FORTRAN DOES NOT PUT ZEROS IN THERE AUTOMATIALLY ALL KASET2(AMATRX, NDOFEL, NDOFEL) IF (NHRS.EQ.1) THEN ALL KASET1(RHS, MLVARX) ELSE ALL KASET2(RHS, MLVARX, NRHS) END IF ALL KASET2(PSI, 6, NDOFEL) 37

39 ALL KASET2(H, MRD, 6) ALL KASET2(AV_OOR, MRD, 3) ALL KASET1(V_XI, MRD) ALL KASET1(V_N, MRD) ALL KASET2(THETA, MRD, MRD) ALL KASET2(DDSDDR, MRD, MRD) ALL KASET2(D_GLOB, MRD, MRD) ALL KASET1(STRESS, MRD) ALL KASET1(STR_GLOB, MRD) REAL INPUT PROPERTIES WIDTH = PROPS(7)! Width of elements (same as solid setion width for solid elements) INTEGER INPUT PROPERTIES NINTP = JPROPS(1)! Number of integration points INTS = JPROPS(2)! Integration point sheme (1: gauss, 2: newton otes) INFORMATION OUTPUT AND HEK IF (iuel.eq.0) THEN OPEN(15,FILE= 1 /user/feih/abaqus/uel/verify.out ) write(7,*) First all to UEL WRITE(7,*) DEGREES OF FREEDOM:,NDOFEL write(7,*) number of nodes:, NNODE write(7,*) number of integration points:, NINTP write(7,*) Integration sheme:, INTS write(7,*) maximum oords:, MRD write(7,*) number of variables:, NSVARS write(7,*) number of real properties, NPROPS write(7,*) number of integer properties, NJPROP write(7,*) dimensioning parameter:, MLVARX write(7,*) KIN:, KIN write(7,*) LFLAGS(1)=, LFLAGS(1) write(7,*) LFLAGS(2)=, LFLAGS(2) write(7,*) LFLAGS(3)=, LFLAGS(3) write(7,*) LFLAGS(4)=, LFLAGS(4) write(7,*) LFLAGS(5)=, LFLAGS(5) HEKING FOR THE RIGHT NUMBER OF NODES IF (NNODE.NE.6) THEN ALL STDB_ABQERR(-3, 6 nodes required for interfae element: 1 speified number of nodes is inorret,0,0.0, ) END IF heking for number of state variables minnum = NINTP*5 IF (NSVARS.LT.minnum) THEN ALL STDB_ABQERR(-3, Number of state variables too small for 1 hosen number of integration points!,minnum,0.0, ) END IF IUEL = 1 END IF WRITE(7,*) New all to UEL do k=1, NDOFEL write(7,*) U, U(k) end do 38

40 all flush_(15) alulate relation matrix alulate relative opening at node pairs DEFINE DELTA_U=U_TOP - U_BOTTOM DO 10 K = 1, NDOFEL/2 PSI(K, K) = -ONE PSI(K, K+NDOFEL/2) = ONE 10 END DO ompute nodal oordinates in deformed state ADD PROPER OORDINATE TRANSFORMATION LATER DO 20 I=1,MRD DO 30 J=1, NNODE NN=I+(J-1)*MRD _OOR(I,J)=OORDS(I,J) + U(NN) 30 END DO 20 END DO Referene oordinate system (midpoint averages) DO 31 I=1, MRD DO 32 J=1, NNODE/2 AV_OOR(I,J)=ONE/TWO*(_OOR(I,J)+_OOR(I,J+NNODE/2)) 32 END DO 31 END DO Gaussian integration (3 gauss points) GAUSS3(1) = -SQRT(0.6) GAUSS3(2) = ZERO GAUSS3(3) = SQRT(0.6) WEIGHT3(1) = WEIGHT3(2) = WEIGHT3(3) = Gaussian integration (6 gauss points) GAUSS6(1) = GAUSS6(2) = GAUSS6(3) = GAUSS6(4) = GAUSS6(5) = GAUSS6(6) = WEIGHT6(1) = WEIGHT6(2) = WEIGHT6(3) = WEIGHT6(4) = WEIGHT6(5) = WEIGHT6(6) = Gaussian integration (12 gauss points) GAUSS12(1) = GAUSS12(2) = GAUSS12(3) = GAUSS12(4) = GAUSS12(5) = GAUSS12(6) = GAUSS12(7) = GAUSS12(8) =

41 GAUSS12(9) = GAUSS12(10) = GAUSS12(11) = GAUSS12(12) = WEIGHT12(1) = WEIGHT12(2) = WEIGHT12(3) = WEIGHT12(4) = WEIGHT12(5) = WEIGHT12(6) = WEIGHT12(7) = WEIGHT12(8) = WEIGHT12(9) = WEIGHT12(10) = WEIGHT12(11) = WEIGHT12(12) = Newton otes integration (3 integration points) OTNEW(1) = -ONE OTNEW(2) = ZERO OTNEW(3) = ONE WEIGHT(1) = ONE/THREE WEIGHT(2) = ONE + ONE/THREE WEIGHT(3) = ONE/THREE Newton otes integration (6 integration points) OTNEW6(1) = -ONE OTNEW6(2) = -3.0d0/5.0d0 OTNEW6(3) = -1.0d0/5.0d0 OTNEW6(4) = 1.0d0/5.0d0 OTNEW6(5) = 3.0d0/5.0d0 OTNEW6(6) = ONE W6(1) = 19.0d0/144.0d0 W6(2) = 75.0d0/144.0d0 W6(3) = 50.0d0/144.0d0 W6(4) = 50.0d0/144.0d0 W6(5) = 75.0d0/144.0d0 W6(6) = 19.0d0/144.0d0 Newton otes integration (12 integration points) OTNEW12(1) = -ONE OTNEW12(2) = -9.0d0/11.0d0 OTNEW12(3) = -7.0d0/11.0d0 OTNEW12(4) = -5.0d0/11.0d0 OTNEW12(5) = -3.0d0/11.0d0 OTNEW12(6) = -1.0d0/11.0d0 OTNEW12(7) = 1.0d0/11.0d0 OTNEW12(8) = 3.0d0/11.0d0 OTNEW12(9) = 5.0d0/11.0d0 OTNEW12(10) = 7.0d0/11.0d0 OTNEW12(11) = 9.0d0/11.0d0 OTNEW12(12) = ONE W12(1) = d0/ d0 W12(2) = d0/ d0 W12(3) = d0/ d0 W12(4) = d0/ d0 W12(5) = d0/ d0 W12(6) = d0/ d0 W12(7) = d0/ d0 W12(8) = d0/ d0 W12(9) = d0/ d0 W12(10) = d0/ d0 W12(11) = d0/ d0 W12(12) = d0/ d0 40

42 IF (LFLAGS(3).EQ.1) THEN Normal inrementation (RHS and AMATRX required) IF (LFLAGS(1).EQ.1.OR.LFLAGS(1).EQ.2) THEN *STATI AND *STATI, DIRET LOOP OVER INTEGRATION POINTS DO 100 IINTP = 1,NINTP IF (NINTP.EQ.3.AND.INTS.EQ.1) THEN POINT = GAUSS3(IINTP) WEIGHT = WEIGHT3(IINTP) ELSE IF (NINTP.EQ.6.AND.INTS.EQ.1) THEN POINT = GAUSS6(IINTP) WEIGHT = WEIGHT6(IINTP) ELSE IF (NINTP.EQ.12.AND.INTS.EQ.1) THEN POINT = GAUSS12(IINTP) WEIGHT = WEIGHT12(IINTP) ELSE IF (NINTP.EQ.3.AND.INTS.EQ.2) THEN POINT = OTNEW(IINTP) WEIGHT = WEIGHT(IINTP) ELSE IF (NINTP.EQ.6.AND.INTS.EQ.2) THEN POINT = OTNEW6(IINTP) WEIGHT = W6(IINTP) ELSE IF (NINTP.EQ.12.AND.INTS.EQ.2) THEN POINT = OTNEW12(IINTP) WEIGHT = W12(IINTP) ELSE WRITE(7,*) Unspeified integration required ALL FLUSH_(7) ALL XIT END IF Shape funtion value H1 = ONE/TWO*(-POINT + POINT**TWO) H2 = ONE/TWO*( POINT + POINT**TWO) H3 = ONE - POINT**TWO DERIVATIVE OF SHAPE FUNTION VALUE (3X1 MATRIX) DNDXI(1) = -ONE/TWO + POINT DNDXI(2) = ONE/TWO + POINT DNDXI(3) = -TWO*POINT H matrix H(1,1) = H1 H(2,2) = H1 H(1,3) = H2 H(2,4) = H2 H(1,5) = H3 H(2,6) = H3 write(7,*) Starting loop over integration points 41

43 write(7,*) INTP POINT and WEIGHT, IINTP, POINT, WEIGHT all flush_(7) ALL KASET2(B, MRD, NDOFEL) DO 110 I=1, MRD DO 120 J=1, NDOFEL DO 130 K=1, NDOFEL/2 B(I,J) = B(I,J) + H(I,K)*PSI(K,J) 130 END DO 120 END DO 110 END DO TRANSPOSED B MATRIX DO 140 I=1, MRD DO 150 J=1, NDOFEL BT(J,I) = B(I,J) 150 END DO 140 END DO ALULATE GLOBAL DISPLAEMENT AT INTEGRATION POINT FROM ONTINUOUS DISPLAEMENT ALL KASET1(DU_ONT, MRD) DO 160 I=1, MRD DO 170 J=1, NDOFEL DU_ONT(I) = DU_ONT(I) + B(I,J)*U(J) 170 END DO 160 END DO LOAL OORDINATE SYSTEM (USE AVERAGE OF DEFORMED X-POSITIONS OF TOP AND BOTTOM) X_xi = ZERO Y_xi = ZERO DO 180 I=1,3 X_xi = X_xi + 1 DNDXI(I)*AV_OOR(1,I) Y_xi = Y_xi + 1 DNDXI(I)*AV_OOR(2,I) 180 END DO Jaobian (vetor length in xi-diretion) DETJ = sqrt(x_xi**two + Y_xi**TWO) IF (DETJ.LT.ZERO) THEN write(7,*) Negative Jaobian enountered! 1 hek element and nodal definition for elem, JELEM ALL XIT END IF Loal oordinate vetor V_XI(1) = X_XI/DETJ V_XI(2) = Y_XI/DETJ Normal vetor in 90 degree angle V_N(1) = - V_XI(2) V_N(2) = V_XI(1) 42

44 Rotational matrix THETA(1,1) = V_XI(1) THETA(2,1) = V_XI(2) THETA(1,2) = V_N(1) THETA(2,2) = V_N(2) Relative displaement in loal oordinate system ALL KASET1(DU_LO, MRD) DO 181 I=1, MRD DO 182 J=1, MRD DU_LO(I) = DU_LO(I) + THETA(J,I)*DU_ONT(J) 182 END DO 181 END DO over-losure hek (an be used as re-start riterion - see uinter) IF (DU_LO(2).LT.TOL) THEN write(7,*) Over-losure at element, JELEM END IF write (7,*) DU_LO:, DU_LO(1), DU_LO(2), IINTP ALL FLUSH_(7) ALULATE STRESS AND TRATION STIFFNESS BASED ON RELATIVE DISPLAEMENT ALL KTRAN(DU_LO, PROPS, STRESS, DDSDDR, 1 MRD, SVARS, NSVARS, IINTP, NINTP, KIN, JELEM) dummy stiffness for frition (no influene under mode I opening when oupled with equation) for auray there should be oupling terms, but again: no influene under mode I opening DDSDDR(1,1) = RHS ASSEMBLY HEK FOR APPLIED LOADS ON STRUTURE IF (NDLOAD.NE.0) THEN WRITE(7,*) Element loads not implemented ALL FLUSH_(7) ALL XIT END IF Stiffness matrix Transformation ALL KASET2(DD1, MRD, MRD) DO 183 I=1, MRD DO 184 J=1, MRD DO 185 K=1, MRD DD1(I,J) = DD1(I,J) + DDSDDR(I,K)*THETA(J,K) 185 END DO 184 END DO 183 END DO ALL KASET2(D_GLOB, MRD, MRD) DO 186 I=1, MRD DO 187 J=1, MRD DO 188 K=1, MRD D_GLOB(I,J) = D_GLOB(I,J) + THETA(I,K)*DD1(K,J) 188 END DO 187 END DO 186 END DO 43

45 ALL KASET2 (A1, NDOFEL, MRD) DO 190 I=1, NDOFEL DO 191 J=1, MRD DO 192 K=1, MRD A1(I,J) = A1(I,J) + BT(I,K)*D_GLOB(K,J) 192 END DO 191 END DO 190 END DO ALL KASET2 (A2, NDOFEL, NDOFEL) DO 195 I=1, NDOFEL DO 196 J=1, NDOFEL DO 197 K=1, MRD A2(I,J) = A2(I,J) + A1(I,K)*B(K,J) 197 END DO 196 END DO 195 END DO DO 200 I=1, NDOFEL DO 201 J=1, NDOFEL AMATRX(I,J) = AMATRX(I,J) + 1 WIDTH*WEIGHT*DETJ*A2(I,J) 201 END DO 200 END DO Right hand side Transformation ALL KASET1(STR_GLOB, MRD) DO 202 I=1, MRD DO 203 J=1, MRD STR_GLOB(I) = STR_GLOB(I) + THETA(I,J)*STRESS(J) 203 END DO 202 END DO DO 230 I=1, NDOFEL DO 240 K=1,MRD RHS(I,1) = RHS(I,1) + 1 DETJ*WIDTH*WEIGHT*BT(I,K)*STR_GLOB(K) 240 END DO 230 END DO IF (NRHS.EQ.2) THEN WRITE(7,*) Riks solution not supported by element ALL FLUSH_(7) ALL XIT END IF IF (LFLAGS(4).EQ.1) THEN PERTURBATION STEP WRITE(7,*) Perturbation step not supported by element ALL FLUSH_(7) 44

46 ALL XIT END IF SAVE OPENING AND STRESSES AT INTEGRATION POINT AS STATE VARIABLES SVARS(IINTP+NINTP) = DU_LO(1) SVARS(IINTP+2*NINTP) = DU_LO(2) SVARS(IINTP+3*NINTP) = STRESS(1) SVARS(IINTP+4*NINTP) = STRESS(2) 100 END DO ELSE WRITE(7,*) Only stati proedure supported by element ALL FLUSH_(7) ALL XIT END IF ELSE IF (LFLAGS(3).EQ.4) THEN DO I=1, NDOFEL AMATRX(I,I)= 1.0d0 END DO ELSE WRITE(7,*) Only normal inrementation supported by element ALL FLUSH_(7) ALL XIT END IF RETURN END subroutine ktran(rdisp, PROPS, STRESS, DDSDDR, MRD, SVARS, 1 NSVARS, IINTP, NINTP, KIN, JELEM) INLUDE ABA_PARAM.IN PARAMETER (ZERO = 0.D0, TWO=2.0D0, ONE= 1.0D0, THREE= 3.0d0) DIMENSION PROPS(*), RDISP(MRD), STRESS(MRD), DDSDDR(MRD, MRD) DIMENSION SVARS(NSVARS) data ifirst/0/ data iopen/0/ data ilose/0/ save ifirst, nodefirst, iopen, ilose REAL INPUT PROPERTIES djss = props(1)!inrease in frature toughness delta = props(2)!max rak bridging opening delta1 = props(3)!initial linear derease/inrease, softening afterwards dj0 = props(4)!value of J0 (from measurements) fa1 = props(5)!stress inrease fator for power law penalty = props(6)!penalty fator on ontat sigma0 = 1.5d0*dJss/delta1*sqrt(delta1/delta) slope1 = -djss/(delta1*delta*sqrt(delta1/delta)) fa = djss/(two*sqrt(delta)) sigma1 = fa/sqrt(delta1) slope = sigma1/delta1 J0 is inluded separately now with zero start power law 45

47 alpha = delta2 = dj0/(sigma0*fa1)*(alpha+1)/alpha ode heks for hange in opening status. If too many ontat points hange status, inrement an be restarted heks for opening/losing behaviour get values from state variables iold = SVARS(IINTP) stressold = SVARS(IINTP + 4*NINTP) rdispold = SVARS(IINTP + 2*NINTP) ode heks for hange in ontat status. If too many ontat points hange status, inrement an be restarted heks for opening/losing behaviour IF (ifirst.eq.0) THEN ifirst = 1 NODEFIRST = NODE write(7,*) djss, delta, delta1, dj0, fa1, penalty write(7,*) sigma0, sigma1, slope, slope1, delta2, fa END IF new inrement detetion (inludes restart) to ount ontat hanges IF (NODE.EQ.NODEFIRST.AND.KIT.EQ.1) THEN iopen = 0 ilose = 0 END IF hek for inreasing opening displaement (start in inrement 2, one all ontat points are losed) Only applies in opening stage (rdisp < 0) Not inluded right now (KIN.GE.1000) IF (rdisp(2).lt.rdispold.and.rdisp(2).gt.delta2.and. 1 KIN.GE.1000) THEN Elasti unloading and reloading stress(2) = stressold/rdispold*rdisp(2) ddsddr(2,2) = stressold/rdispold write(7,*) Elasti unloading enountered ELSE hek for penetration of surfaes and indiate status if (rdisp(2).lt.zero) then write(7,*) Area I stress(2) = penalty*slope*rdisp(2) ddsddr(2,2) = penalty*slope lopenlose = 0 hek for opening of rak Stresses will be negative (tension) First slope bit (different from square root law) else if (rdisp(2).ge.zero.and.rdisp(2).lt.delta2) then Initial inrease write(7,*) Area II stress(2) = fa1*sigma0* 1 (1.0-((delta2-rdisp(2))/delta2)**alpha) ddsddr(2,2) = fa1*alpha*sigma0/delta2* 1 (((delta2-rdisp(2))/delta2)**(alpha-1)) lopenlose = 1 46

48 Softening behaviour else if (rdisp(2).ge.delta2.and. 1 rdisp(2).lt.(delta1+delta2)) then write(7,*) Area III stress(2) = sigma0 + slope1*(rdisp(2)-delta2) ddsddr(2,2) = slope1 lopenlose = 2 else if (rdisp(2).ge.(delta1+delta2). 1 and.rdisp(2).lt.(delta+delta2)) then write(7,*) Area IV stress(2) = fa/sqrt((rdisp(2)-delta2)) ddsddr(2,2) = -fa/2*((rdisp(2)-delta2))**(-3.d0/2.d0) lopenlose = 3 else if (rdisp(2).gt.(delta+delta2)) then write(7,*) Area V stress(2) = 0 ddsddr(2,2) = 0 lopenlose = 4 end if END IF IF (lopenlose.ne.iold.and.iold.eq.0.and.kin.gt.3) THEN Restart if more than one ontat pair opens (iold=0) in 2nd all IF (KIT.EQ.2) THEN iopen = iopen + 1 END IF write(7,*) Status: iopen=, iopen, at int point, IINTP, 1 in element, JELEM, and inrement, KIN write(7,*) lopenlose =,lopenlose, iold=,iold, KIT=, KIT Possible restart proedure IF (iopen.gt.1) THEN write(7,*) Too many ontat openings: redue inrement END IF END IF Restart if one ontat pair loses (iold=1/lopenlose=0) IF (lopenlose.eq.0.and.iold.eq.1.and.kin.ge.2) THEN ilose = ilose + 1 write(7,*) Status: ilose=, ilose, at int point, IINTP, 1 in element, JELEM, and inrement, KIN write(7,*) lopenlose =,lopenlose, iold=,iold, KIT=, KIT IF (ilose.gt.0) THEN write(7,*) Elasti unloading possible: redue inrement END IF END IF Restart with PNEWDT (if PNEWDT less than 1) IF (ilose.gt.0) THEN PNEWDT = 1.0 ELSE IF (iopen.gt.4) THEN PNEWDT =

49 END IF Sign definition (bridging stress ats as losure stress on struture as in ontat analysis) Stiffness matrix aording to ABAQUS definition: -df/du!! stress(2) = - stress(2) State variable update SVARS(IINTP) = lopenlose return end subroutine KASET1(DMATRIX, IDIMX) INLUDE ABA_PARAM.IN PARAMETER (ZERO = 0.0D0) DIMENSION DMATRIX(IDIMX) DO i=1, IDIMX DMATRIX(i) = ZERO END DO RETURN END subroutine KASET2(DMATRIX, IDIMX, IDIMY) INLUDE ABA_PARAM.IN PARAMETER (ZERO = 0.0D0) DIMENSION DMATRIX(IDIMX, IDIMY) DO I = 1, IDIMX DO J = 1, IDIMY DMATRIX(I,J) = ZERO END DO END DO RETURN END

50 E ohesive element verifiation E.1 Verifiation of the 2D element Figure 24 shows the simple two element model omparing the previously written subroutine UINTER [14] and the new user element. Both models give idential results. The nodes 1-6 for the user element initially oinide in their position: here they are plotted apart for a better understanding of the node numbering Slave surfae Master surfae (a) 2D ontat model (b) 2D user element verifiation Figure 24. 2D user element model The input dek is simple and given in the following for the two-element problem. Note that the element is under plane stress ondition (element, type=ps8) to give omparable results to the 3D model with its boundary onditions. *HEADING *NODE 1, 0.0, 0.0 2, 1.0, 0.0 3, 0.5, 0.0 4, 0.0, 0.0 5, 1.0, 0.0 6, 0.5, 0.0 7, 1.0, 1.0 8, 0.0, 1.0 9, 1.0, , 0.5, , 0.0, 0.5 *NSET, NSET=ALL 8, 10, 9 *user element, type=u6, nodes=6, oordinates=2, i properties=2, properties=7, variables=15 1, 2 *ELEMENT, TYPE=U6, ELSET=ALL 1, 1, 2, 3, 4, 5, 6 *ELEMENT, TYPE=PS8, ELSET=eall 2, 4, 5, 7, 8, 6, 9, 10, 11 *SOLID SETION, ELSET=EALL, MATERIAL=MAT1, ORIENTATION=ORIENT1 1.0 *UEL PROPERTY, ELSET=ALL 1.65, 1.0, 0.5, 0.15, 1.0, , 1.0, 3 1 *MATERIAL, name=mat1 *ELASTI, TYPE=ENGINEERING ONSTANTS 41.5E3, 9.5E3, 9.5E3, 0.3, 0.3, 0.3, 15.8E3, 3.65E3 3.65E3 49

51 *orientation, name=orient1 1.0, 0.0, 0.0, 0.0, 1.0, 0.0 1, 0. *BOUNDARY 1, 1, 2 2, 2, 2 3, 2, 2 4, 1, 1 11, 1, 1 8, 1, 1 *STEP, IN=100, NLGEOM *STATI 0.01, 1.0,,0.01 *MONITOR, DOF=2, NODE=4 *ONTROLS, PARAMETERS=TIME INREMENTATION 7, 10, 9, 16, 10, 4, 20, 10, 6 *BOUNDARY 8, 2, 2, 1.0 7, 2, 2, , 2, 2, 1.0 *OUTPUT, FIELD, FREQ=1 *ELEMENT OUTPUT S, E, *NODE OUTPUT U, RF, *OUTPUT, HISTORY, FREQ=1 *NODE OUTPUT, NSET=ALL U2, RF2 *end step Figure 25 shows the results for the test ase as reated by ABAQUS/AE. As the stiffness of the top element is large ompared to the bridging stress input, the stress distribution in loading diretion appears onstant. The displaement field in x-diretion, whih is aused by the Poisson s effet, varies linearly. For the simple ase of uniform deformation (the same displaement boundary ondition on all top nodes), the original bridging law is obtained from the reation fores, element area and the opening of the element. S, S e e e e e e e e e e e e-01 U, U e e e e e e e e e e e e e-06 (a) ontour plots (b) History plot Figure 25. Verifiation of 2D user element 50

52 E.2 Verifiation of the 3D element The input dek is simple and given in the following for the two element test analogous to the 2D model: *HEADING *NODE 1, 0.0, 0.0, 1.0 2, 1.0, 0.0, 1.0 3, 1.0, 0.0, 0.0 4, 0.0, 0.0, 0.0 5, 0.5, 0.0, 1.0 6, 1.0, 0.0, 0.5 7, 0.5, 0.0, 0.0 8, 0.0, 0.0, 0.5 9, 0.0, 0.0, , 1.0, 0.0, , 1.0, 0.0, , 0.0, 0.0, , 0.5, 0.0, , 1.0, 0.0, , 0.5, 0.0, , 0.0, 0.0, , 0.0, 1.0, , 1.0, 1.0, , 1.0, 1.0, , 0.0, 1.0, , 0.5, 1.0, , 1.0, 1.0, , 0.5, 1.0, , 0.0, 1.0, , 0.0, 0.5, , 1.0, 0.5, , 1.0, 0.5, , 0.0, 0.5, 0.0 *NSET, NSET=ALL 8, 10, 9 *NSET, NSET=BOTTOM 1, 2, 3, 4, 5, 6, 7, 8 *NSET, NSET=TOP 17, 18, 19, 20, 21, 22, 23, 24 *NSET, NSET=SIDEX 1, 4, 8, 9, 12, 16, 17, 20, 24, 25, 28 *NSET, NSET=SIDEZ 1, 2, 5, 9, 10, 13, 17, 18, 21, 25, 26 *user element, type=u16, nodes=16, oordinates=3, i properties=2, properties=7, variables=63 1, 2, 3 *ELEMENT, TYPE=U16, ELSET=ALL 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 *ELEMENT, TYPE=3D20, ELSET=eall 2, 9, 10, 11, 12, 17, 18, 19, 20, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 27, 28 *SOLID SETION, ELSET=EALL, MATERIAL=MAT1, ORIENTATION=ORIENT1 *UEL PROPERTY, ELSET=ALL 1.65, 1.0, 0.5, 0.15, 1.0, , 1.0, 3 1 *MATERIAL, name=mat1 *ELASTI, TYPE=ENGINEERING ONSTANTS 41.5E3, 9.5E3, 9.5E3, 0.3, 0.3, 0.3, 15.8E3, 3.65E3 3.65E3 *orientation, name=orient1 1.0, 0.0, 0.0, 0.0, 1.0, 0.0 1, 0. *BOUNDARY BOTTOM, 2,, 0.0 SIDEX, 1,, 0.0 SIDEZ, 3,, 0.0 *STEP, IN=100, NLGEOM 51

53 *STATI 0.01, 1.0, *MONITOR, DOF=2, NODE=9 *ONTROLS, PARAMETERS=TIME INREMENTATION 7, 10, 9, 16, 10, 4, 20, 10, 6 *BOUNDARY TOP, 2, 2, 1.0 *OUTPUT, FIELD, FREQ=1 *ELEMENT OUTPUT S, E, *NODE OUTPUT U, RF, *OUTPUT, HISTORY, FREQ=1 *NODE OUTPUT, NSET=ALL U2, RF2 *end step The stress and displaement field are idential to the results presented for the 2D model in the previous appendix. For the simple ase of uniform deformation (the same displaement boundary ondition on all top nodes), the original bridging law is obtained by summing up all reation fores on the top of the element and reording the opening of the element. S, S e e e e e e e e e e e e e U, U e e e e e e e e e e e e e-06 (a) ontour plots (b) History plot Figure 26. Verifiation of 3D user element 52