Akkerman, Remko

ten Thije, R.H.W.

English

Intra-ply shear locking results in unrealistic fibre stresses and spurious wrinkling in composite forming simulations. Three remedies are investigated: aligning the mesh, applying reduced integration and using multi-field elements. The bias extension simulation is used to test several triangular and quadrilateral elements on their capability to avoid locking. Their performance under large deformations is tested as well. The new multi-field element seems to be the best locking free element in random meshes

University of Twente Research Information

Intra-ply shear lockingR.H.W. ten Thije and R. AkkermanDepartment of Mechanical Engineering, University of Twente, P.O. Box 217, 7500 AE Enschede,The Netherlands. tel +31(0)53 489 2426, email: r.h.w.tenthije@utwente.nlAbstract. Intra-ply shear locking results in unrealistic fibre stresses and spurious wrinkling in com-posite forming simulations. Three remedies are investigated: aligning the mesh, applying reducedintegration and using multi-field elements. The bias extension imulation is used to test several tri-angular and quadrilateral elements on their capability to avoid locking. Their performance underlarge deformations is tested as well. The new multi-field elem nt seems to be the best locking freeelement in random meshes.Keywords: intra-ply shear locking, finite element, textiles, formingPACS: 02.70.DhINTRODUCTIONManufacturing processes can lead to unacceptable shape distortions in fibre reinforcedproducts. These distortions depend on a wide variety of parameters from which geom-etry, material properties, lay-up, process temperatures and friction are just a few of theimportant factors. Numerical tools such as finite element (FE) simulations are a power-ful tool to analyse these distortions and in the ideal case can le d to a first-time-rightdesign. These optimizations require a robust, accurate andyet fast numerical procedure.This is not evident for anisotropic materials as shown in [1]. Besides this problem thereis another important problem that has to be solved: intra-ply shear locking.Intra-ply shear lockingIntra-ply shear locking is nothing more than the incapability of the element displace-ment field to represent the correct deformation mechanism ofthe fibre reinforced mate-rial. It is a numerical problem that has nothing to do with thephysical jamming of thereinforcement. It leads to overestimation of fibre stresses, forces and stiffness and oftenleads to spurious wrinkles in 3D simulations. Some authors already suspected the exis-tence of a locking phenomena, but the first article that addresses the problem correctlyand illustrates the shortcomings of the current standard elem nt formulation was pre-sented by Yu et al. [2]. The locking phenomena can be illustrated by simulating the biasextension experiment. The bias extension experiment is frequently used to examine theshear response of biaxial reinforced materials. Figure 1a show the undeformed and thedeformed shape of the material. The two fibre directions are initially perpendicular toeach other at±45◦. The specimen is gripped on the short edges and gradually extended.The stiffness of the fibres is dominant and the material deforms as a trellis frame, witheach fibre crossing acting as a possible hinge point. Three deformation regions develop:an undeformed region (I), a central region with pure shear (II) and a region with inter-mediate shear (III). The paper of Potter [3] describes the bias extension experiment indetail. The finite element borders are aligned with the fibre di ctions in figure 1a and(a) Mesh A (aligned)1011021031040102030405060708090normalized force [N/N]degrees of freedom [-]mesh Amesh BI(b) Mesh B (unaligned)(c)IIIIIIIIIIIFigure 1. The three different deformation regions in a bias experiment: I, II and III. The aligned mesh (a)does not lock and the unaligned mesh (b) locks as illustratedin figure (c).coincide with the lines along which the material develops ’hinge’ lines. This mesh canrepresent the trellis deformation correctly and locking isab ent. The mesh shown in fig-ure 1b shows severe locking, since a standard element cannotrepresent the discontinuousshear field. Fibres are stretched during the simulation, resulting in unrealistically highforces. Figure 1c shows the resulting tensile force of a biasextension simulation withboth mesh types. Locking is reduced if the number of elementsincreases. This is the casein most types of locking. It is clear that the excessive amount f elements necessary toeliminate locking for this problem is unacceptable.REMEDIES AGAINST INTRA-PLY SHEAR LOCKINGThree possible ways to avoid locking are discussed in this paper:1. Aligned meshes2. Selective reduced integration3. Multi-field elementsSome of these options make it necessary to use higher order elements or quadrilaterals.Other options such as XFEM are not considered here.Aligned meshes.Aligning the mesh with the fibre directions is probably one ofthebest remedies against intra-ply shear locking. Special attention is needed at curved edgesof the blanks during meshing, where element edges cannot simply follow the curvedboundary. This was already shown in the drape simulation in the article of Yu et al. [2].This limits the use of automatic mesh generation. Alignmentbecomes impossible if theelement contains more than two fibre directions. This can occur in multi-layer elements,where several layers are efficiently modelled within one elem nt through the thickness asin [4]. Alignment also limits the use of Arbitrary Lagrangian Eulerian (ALE) methods,where the alignment of the fibres and the mesh will disappear during the simulation.Selective reduced integration (SRI).Reduced integration or underintegration ofelements is a well known way to eliminate several types of locking. Yu et al. [2] il-lustrated the use of reduced integration to eliminate intra-ply shear locking for wovenfabrics. It was not considered a useful option due to the hourglassing introduced by thezero energy modes. A way to avoid this is to use Selective Reduced Integration (SRI).Reduced integration should only be applied to the fibre contribution of the materialmodel. Using SRI in an element that is (accidentally) alignedwith the fibre directionresults in unnecessary deformation modes. The program becomes more robust if it ischecked whether the element edge is aligned with the fibre or not. Full integration shouldbe used if the element is aligned, otherwise SRI should be used. This process is calledAdaptive Selective Reduced Integration (ASRI).The locking problem cannot be solved for the linear triangleby reduced integration,since it is already fully integrated with only one point. Higher order elements areneeded, e.g. quadratic triangles. The order of the displacement field of an element can beincreased by adding degrees of freedom to existing nodes instead of adding nodes. Anexample is the Allmann88 triangle with vertex rotations [5,6, 7]. Adding bubble modesor incompatible modes appeared to be ineffective.Multi-field element (MF). Multi-field elements are also known as mixed elements.They contain degrees of freedom of different types, e.g. displacement and temperature.In this case the fibre strain is chosen as an additional degreeof freedom beside the normaldisplacement degrees of freedom. A fibref with directiona is located in a continuumat positionx. The strain ratėε f of the fibre is the degree of freedom and must equal thevelocity gradient in the direction of this fibre:ε̇ f = a·v←∇ ·a (1)wherev denotes the velocity. The weak FE formulation is found by weighing with a func-tion w and integrating over the volume. With the use of the chain rule for differentiationand the divergence theorem of Gauss, the equation can be writt n as:∫Vwε̇ f dV +∫Vw←∇(a·v)a dV =∫Γwn(a·v)a dΓ (2)The natural boundary condition for the strain field is zero velocity of the fibre withrespect to the normal on the surface boundary. The nodes thatcontain the strain degreesof freedom can not be placed on the element edges. Placing theodes on the edges wouldallow fibres to flow from one element into the adjacent element. The multi-field elementcombines strains and displacements within one element. Strain fields can have a differentorder of magnitude compared to displacements. A lot of significant digits can be lost dueto this mismatch, making simulations less robust and less accur te. Another disadvantageis that the element stiffness matrix of the multi-field element can become non positivedefinite. This can be a problem for solvers used in FE calculations.IMPLEMENTATION AND RESULTSTen different elements are selected based on the remedies against intra-ply shear lockingthat were discussed in the previous section. They are listedbelow and shown in figure 2.1. Simplex (linear) triangle (LTR)2. Quadratic triangle (QTR)3. Quadratic triangle with selective reduced integration (QTR_SRI)4. Quadratic triangle with adaptive selective reduced integration (QTR_ASRI)5. Allman88 triangle (DRIL)6. Allman88 triangle with selective reduced integration (DRIL_SRI)7. Multi-field quadratic triangle with a constant strain field (QTR_MF)8. Simplex (linear) quadrilateral (QUAD)9. Simplex (linear) quadrilateral with selective reduced integration (QUAD_SRI)10. Multi-field simplex quadrilateral with a constant strain field (QUAD_MF)displacementmultifieldrotationalDegrees of freedomfullreducedIntegration schemeLTR QTR / QTR_SRIQTR_ASRIDRIL / DRIL_SRIQTR_MF QUAD / QUAD_SRI QUAD_MFFigure 2. Ten different elements that are tested on their performancein a bias extension simulation.Table 1. Parameters of the bias extension simulation.width [mm] 50 bulk stiffness [MPa] 2.0hight [mm] 100 fibre stiffness [MPa] 4000.0thickness [mm] 1 bulk volume fraction [-] 0.50mesh [-] 2 x 4 fibre volume fraction (2x) [-] 0.25The elements were tested in a bias extension simulation as shown in figure 1 in arelatively coarse mesh. The number of elements ranged between 8 and 16, depending onthe element type and mesh type. The geometry and the materialpa meters of the biasextension simulation can be found in table 1. Aligned and unalig ed meshes of triangleswere used, whereas the mesh of the quadrilaterals was necessarily unaligned. Figure 3shows the resulting tensile forces of the simulations. The tensile forces are normalisedwith respect to the result of the simulation with simplex, aligned triangles. The fullyintegrated elements show accurate results in the aligned mesh and show severe lockingin the unaligned meshes as expected. The quadratic triangle(QTR) gives quite accurateresults when used in a random mesh with an overshoot of only 20%. Apparently, thecurved boundaries of this element can conform to the discontinuous fields quite wellin this case, were the elongation of the specimen is only 1%. The QTR shows severelocking if the elongation is increased. The results from themulti-field (MF) elements andthe elements with reduced integration (SRI) are almost identcal. Reduced integrationreduces the number of evaluated fibre strains in an element toone, equal to the number0.850.90.951DRIL1.01LTR1.00QTR_ASRI0.99QTR0.99QTR_MF0.91QTR_SRI0.91normalized force [N/N]QUAD22.79LTR8.23DRIL_SRI2.92QTR1.20QUAD_MF0.97QUAD_SRI0.96QTR_MF0.87QTR_SRI0.87aligned mesh random meshFigure 3. Tensile forces from simulation of the bias extension simulation. The forces are normalised withrespect to the simulations with aligned simplex triangles.0246810121416182022QTR_MFLTRQTR_ASRIQTRQTR_SRIDRIL_SRIMaximum displacement increment [mm]QTR_MFQUAD_MFQUAD_SRIQTR_ASRIQTR_SRIaligned mesh random meshFigure 4. Maximum incremental displacement within one step. Testingthe performance of the elementsin large deformations.of fibre strains in the multi-field element. It seems to be an effective way to eliminatelocking. Elements become significantly less stiff, even less stiff than the aligned, fullyintegrated elements. This effect is known from other types of locking: reduced integration’over’-eliminates the locking effect. The Allman88 element (DRIL) with drilling degreesof freedom and with SRI still locks in a random mesh. It seems that at least a quadraticfield with six additional degrees of freedom is needed to avoid l cking in triangularelements. Forming simulations of fibre reinforced composites must perform well withlarge deformations of highly anisotropic material. The locking free elements from figure3 are tested on their performance in large deformations. Thefibre stiffness is increased bya factor 103 and the maximum incremental displacement of one step is determin d. Thetangent matrix is calculated numerically for all element types and the maximum numberof iterations is set to 8. Figure 4 shows the results of this analysis. The performanceof elements deteriorates when reduced integration is used and improves for multi-fieldelements. The maximum step size for the aligned quadratic triangles is around 7% ofthe total specimens length. This value increases by almost 2.5 times for the multi-fieldelement and decreases by almost 2.5 times for the element that uses SRI. For randommeshes the difference is even larger. The quadrilaterals perform quite well and thedifference between SRI and the multi-field element is not as large s for the quadratictriangle. The performance of the Allman88 triangle is poor.CONCLUSIONS AND FUTURE WORKThree remedies against intra-ply shear locking are investigated in this paper: aligning themesh, applying selective reduced integration (SRI) and using multi-field elements. Allfully integrated standard elements have to be aligned with the fibre directions to avoidlocking. For a random mesh one can choose a higher order triangle (at least quadratic)with SRI or a simplex quadrilateral (or higher) with SRI. SRI indicates that only the fibrecontributions to the material model are underintegrated. The new multi-field elementscombine the standard displacement degrees of freedom with an additional field that in-terpolates the fibre strain. The elements are closely related to the SRI element and arefree of locking for any mesh configuration. Aligning the meshi the best option withrespect to accuracy, but limits the use of automatic mesh generation and makes imple-mentation of multi-layer elements impossible. Elements baed on SRI and multi-fieldelements underestimate the stiffness, typically from 5% down to 15% in this particularstudy. The maximum allowable incremental step size during asimulation is much largerfor the multi-field element than for the elements using SRI andeven larger compared tothe maximum step size of an aligned element. The multi-field elem nt is the best elementto use in large deformations of fibre reinforced materials inrandom meshes, based on theresults from this analysis. The next step is the implementation of the multi-field elementin a 3D simulation to verify the performance in realistic 3D simulations.REFERENCES1. R.H.W. ten Thije, R. Akkerman, J. Huétink,Large deformation simulation of anisotropic materialPr.9th ESAFORM Conf., Publ. House Akapit, Krakow, Poland, 2006, p. 803-806, ISBN 83-89541-66-1.2. Xiaobo Yu, Bruce Cartwright, Damian McGuckin, Lin Ye and Yiu-Wing Mai, Intra-ply shear lockingin finite element analyses of woven fabric forming processes. Composites Part A: Applied Science andManufacturing, Volume 37, Issue 5, May 2006, p. 790-803.3. Kevin Potter,Bias extension measurements on cross-plied unidirectional prepreg, Composites Part A:Applied Science and Manufacturing, Volume 33, Issue 1, January 2002, Pages 63-73.4. R.H.W. ten Thije, R. Loendersloot, R. Akkerman,Drape simulation of non-crimp fabrics, Proc. 8thInt. ESAFORM conf., Publishing house of Romanian Academy, Bucharest, Romania, 2005, Pages991-994, ISBN 973-27-1174-6.5. D.J. Allman,A Compatible Triangular Element Including Vertex rotations for plane elasticity analysis,Computers & Structures, Volume 19, Issues 1-2, 1984, Pages 1-8.6. D.J. Allman,Evaluation of the constant strain triangle with drilling rotations, Int. J. Numer. Meth.Engrg., Volume 26, 1988, Pages 2645-2655.7. Carlos A. Felippa,A study of optimal membrane triangles with drilling freedoms, Computer Methodsin Applied Mechanics and Engineering, Volume 192, Issues 16-18, 25 April 2003, Pages 2125-2168.

Intra-ply shear locking

https://ris.utwente.nl/ws/files/5403994/Thije07intra.pdf

Intra-ply shear locking

Abstract

Similar works

Full text

Available Versions

University of Twente Research Information