This paper presents a numerical methodology to model elastic braided fibres. Elastic transversely isotropic material was used due to the anisotropy of the strands. Large deformation finite analysis based on large rotations / small strains total Lagrangian formulation was used to account for geometrical nonlinearities. Pre-processing was necessary to insert interface elements where strands are in self contact and with other materials, followed by a potential flow analysis that was undertaken to evaluate the fibre directions for every individual strand. An FE model of a knot demonstrates the important of defining interface elements, while a three plait braided model shows the importance of using a transversely isotropic model compared to an isotropic model. A twelve strand sinnet rope (T12) was also modelled using the proposed methodology and highlights new challenges when using dierent types of braiding

Cortis, M.

Gil, Antonio J.

Kaczmarczyk, L.

Pearce, C.

Sevilla, Rubén

English

Durham Research Online

Proceedings of the 23rd UK Conference of theAssociation for Computational Mechanics in Engineering8 - 10 April 2015, Swansea University, SwanseaFinite element modelling of braided fibres subject to large deformations*M. Cortis1, L. Kaczmarczyk 2 and C.J. Pearce31School of Engineering and Computing Science, Durham University, Durham, DH1 3LE2,3School of Engineering, University of Glasgow, Rankine Building, Glasgow, G12 8LT*michael.cortis@durham.ac.ukABSTRACTThis paper presents a numerical methodology to model elastic braided fibres. Elastic transverselyisotropic material was used due to the anisotropy of the strands. Large deformation finite analysis basedon large rotations / small strains total Lagrangian formulation was used to account for geometrical non-linearities. Pre-processing was necessary to insert interface elements where strands are in self contactand with other materials, followed by a potential flow analysis that was undertaken to evaluate the fibredirections for every individual strand. An FE model of a knot demonstrates the important of defining in-terface elements, while a three plait braided model shows the importance of using a transversely isotropicmodel compared to an isotropic model. A twelve strand sinnet rope (T12) was also modelled using theproposed methodology and highlights new challenges when using di↵erent types of braiding.Key Words: fibre reinforced concrete; braided ropes; knots; large deformation; transversely isotropic;interface elements1. IntroductionBraided sinnet ropes made from synthetic materials such as carbon and Technora fibre can be used asflexural strengthening of concrete structures but it is essential that their mechanical behaviour and inter-action with other materials is well understood. A system was developed, using technora T12 rope braidedover a series of glass beads, creating ribbing which increases the adhesion and pull-out strength whenembedded in concrete [1]. This continuous braided fibre reinforcement system could eventually replaceFRPs reinforcement and producing fire resistant concrete structures. The resin matrix of FRPs have alow glass transition temperature (GTT) of around 200 C, loosing its composition and bond strength by80-90% [3, 4]; in comparison, continuous braided fibre reinforcement systems make use of braiding tobind fibres and mitigate problems related to polymeric resins.In this paper we propose a finite element modelling approach to simulate the mechanical behaviour ofelastic braided fibres subject to large deformations. The importance of such technique is demonstratedby modelling a knot and a three braided plait under tension.2. Large Deformation for transverse isotropic materialBraided geometries can experience two types of geometric non-linearities, large material rotations dueto inter-winding geometries, and large strains when they are made for soft materials. In this paper we useelastically sti↵materials such as carbon and Technora fibres and hence the finite deformation formulationis restricted to large rotations and small strains. The strain energy density function is based on the simplestpotential energy function  = 12 E : D : E for elastic materials, where E is the Green strain tensor and Dis the elastic transversely isotropic sti↵ness matrix. A total Lagrangian formulation was used, althoughupdated Lagrangian formulation is recommended for better convergence of geometric non-linearities.For curved strands it is necessary to automate the process of calculating the fibre orientations in order tocorrectly define the transversely isotropic sti↵ness. To achieve this a steady-state laminar incompressiblepotential flow analysis is undertaken and the gradient of the resulting potential field computed for every405strand [2] to give the fibre orientation. One of the numerical examples below (three plait model undertension) demonstrates the importance of using transverse isotropy and large deformation for such cases.Where strands are in contact, zero thickness interface elements are inserted before the potential field iscomputed, as demonstrated in the knot’s example below, Figure 1. Furthermore, the mechanical inter-action between materials can be simulated using cohesive models providing resistance due to tangentialand shear gap opening. Normal penetration was avoided by using a high penalty sti↵ness in the normaldirection [2]. Computation of the fibre orientation and construction of the interface elements was imple-mented as a pre-processing stage in MoFEM (FE code developed at the University of Glasgow) beforemechanical analysis was undertaken.3. Knot ModelTo illustrate the importance of interface elements, and finite deformations, a knot made from a 1 mmdiameter strand was modelled and subjected to stretching. The elastic transversely isotropic materialparameters were Ez = 135GPa and Ep = 50GPa as the principal and transverse sti↵ness of strands,⌫p = 0.4 as the Poisson’s ratio in the transverse plane, and ⌫pz = 0.1 and Gzp =Ez2(1+⌫pz) as the Poisson’sratio and shear modulus between the principal axis and transverse plane respectively. Elastic interfaceelements were considered with penalty sti↵ness of Dn = 1350GPa and Dt1 = Dt2 = 1GPa for the normaland two tangential components. Interface elements were inserted where the knot self touches as shownin Figure 1-left. The approach undertaken was to physically detach any surface in contact and determinefibre orientations (Figure 1-right) from the gradient of the computed potential field having all the zeroflux boundary walls defined. This avoids ‘leakages’ through contact areas (Figure 1-middle).Figure 1: Left: Interface within the knot; Middle: Wrong fibre orientation; Right: Correct fibre orientationThe model was fixed on one end and pulled at an angle of 45  from the other end. Exponential sti↵eningwas observed in the first phase due to straightening, followed by changes in sti↵ness during tighteningand repetitive slipping/re-tightening process of the knot. This is a common mechanism of knots, mainlydependent on the coe cient of friction and the type of knot in use. Large deformation modelling wasessential due to the large rotations experienced - see Figure 2.0200004000060000800000 1 2 3 4 5 6 7 8Displacement [mm]StretchingForce[N]StiffeningofknotSlippage andre-tighteningFigure 2: Axial load-displacement graph for stretching behaviour of knot4. Modelling of Braided GeometriesTwo di↵erent models of braided geometries were investigated in this section. The first model consistedof a three plait geometry (dimensions shown in Figure 5). The second model represented a twelve strands406sinnet (T12) rope clamped between two steel plates (Figure 6). The same material parameters as for theknot example were used, except ⌫p = 0.3 in the T12 model.The three plait model was subject to tension and modelled assuming both small and large deformationtheorem. Small deformation modelling exhibited spurious bending (Figure 3) compared to the largedeformation model (Figure 5) assuming small strains/large rotations. This bending was due to materialswelling and emphasises that geometric non-linearity cannot be ignored.Another test was defined to illustrate the importance of using transverse isotropy when modelling braidedfibrous strands. The same three plait model was analysed using both elastic isotropic (E = 135GPa and⌫ = 0.1) and transversely isotropic material, taking into consideration geometric non-linearities. Artificialtorsion was observed with the maximum twist taking place towards the centre in the isotropic model, asshown by the overlap image in Figure 4.The overall sti↵ening mechanism can be observed in Figure 5, while p-refinement using hierarchicalhigher-order approximations indicate that the solution is converging. A Newton iterative solver witharc-length procedure and secant line search was used to improve the convergence of the geometricallynon-linear problems.The modelling of the T12 sinnet rope was more challenging due to the geometric formation of a rigidlattice structure arrangement with a nearly constant sti↵ness, as shown in Figure 6. It is suggested thatfurther investigation is required as to how di↵erent geometry configurations and material parametersa↵ect the overall mechanism of sinnet braids. Similar to the three plait model, p-refinement showedconvergence but required significant increase in computational resources.A major problem when using transversely isotropic materials is the possibility of sti↵ness matrix ill-conditioning if Ez and Ep are very di↵erent in magnitude. The same problem occurs at the interfacewhen Dn is significantly larger or smaller than Dt1,t2.Figure 3: Un/deformed shape using small deformation Figure 4: Deformed shape (half model) ofIsotropic and Transversely Isotropic using largedeformationPitch = 3.14Dia. = 0.410200040006000800010000120000 0.5 1 1.5 2 2.5 3Pull-outforce[N]Displacement [mm]3P_TI_PO23P_TI_PO33P_TI_PO1Figure 5: Load-displacement graph for 3-plaited model using finite deformation theorem40725 mm25 mmFullyFixedFixedx & y-axiszxy6mm53mm6mmDia.2.88mm0200004000060000800001000001200000 0.5 1 1.5 2Force [N]Displacement [mm]Polynomial Degree 1Polynomial Degree 2Polynomial Degree 3Figure 6: Load-displacement graph for T12 sinnet rope (including p-refinement)5. ConclusionsThis paper presented a numerical methodology to model braided fibres leading to better understanding oftheir overall mechanical behaviour. An elastic transversely isotropic material model was used to representstrands made from materials such as carbon, glass, kevlar or technora. It was essential to consider sti↵nessin the transverse plane of the strands, in particular when modelling braided fibres. Geometric non-linearanalysis was vital due to large rotations during deformation. It was also found that large variations inthe anisotropic sti↵ness properties and di↵erences between the sti↵ness in the normal and tangentialdirections of the interface elements leads to numerical instabilities during pre-conditioning of the globalsti↵ness matrix. The examples have shown the importance of the adopted approach while further studiesare required to demonstrate its applicability on di↵erent types of braiding/weaving and the influence ofdi↵erent material parameters.AcknowledgementsThe research work undertaken in this publication is funded by the Strategic Educational Pathways Schol-arship (Malta). The scholarship is part-financed by the European Union European Social Fund (ESF)under Operational Programme II Cohesion Policy 2007-2013, Empowering People for More Jobs and aBetter Quality of Life.References[1] M.Cortis, L.Kaczmarczyk, C.J. Pearce. Computational modelling and experimental investigation of the bond behaviourbetween concrete and braided fibre ropes, International Conference on Computational Mechanics (CM13), Durham,March 2013[2] M. Cortis, L. Kaczmarczyk, C.J. Pearce. Computational modelling of braided fibre for reinforced concrete, 22nd UKConference of the Association for Computational Mechanics in Engineering (CM14), Exeter, April 2014[3] A. Katz, N. Berman, and L.C. Bank. E↵ect of high temperature on bond strength of FRP rebars. Journal of Compositesfor Construction, 3(2): 73 81, May 1999.[4] A. Katz and N. Berman. Modelling the e↵ect of high temperature on the bond of FRP reinforcing bars to concrete.Cement and Concrete Composites, 22(6):433 443, December 2000.408

Finite element modelling of braided fibres subject to large deformations

https://durham-repository.worktribe.com/file/1153959/1/Published%20Conference%20Proceeding

Finite element modelling of braided fibres subject to large deformations

Abstract

Similar works

Full text

Available Versions

Durham Research Online