mechanical modelling of pultrusion process_ 2d and 3d numerical approaches
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Appl Compos Mater (2015) 22:99118DOI 10.1007/s10443-014-9394-3
Mechanical Modelling of Pultrusion Process: 2D and 3DNumerical Approaches
Ismet Baran Jesper H. Hattel Remko Akkerman Cem C. Tutum
Received: 21 March 2014 / Accepted: 7 April 2014 / Published online: 29 May 2014 Springer Science+Business Media Dordrecht 2014
Abstract The process induced variations such as residual stresses and distortions are a crit-ical issue in pultrusion, since they affect the structural behavior as well as the mechanicalproperties and geometrical precision of the final product. In order to capture and investigatethese variations, a mechanical analysis should be performed. In the present work, the twodimensional (2D) quasi-static plane strain mechanical model for the pultrusion of a thicksquare profile developed by the authors is further improved using generalized plane strainelements. In addition to that, a more advanced 3D thermo-chemical-mechanical analysis iscarried out using 3D quadratic elements which is a novel application for the numerical mod-elling of the pultrusion process. It is found that the 2D mechanical models give relativelyreasonable and accurate stress and displacement evolutions in the transverse direction ascompared to the 3D model. Moreover, the generalized plane strain model predicts the lon-gitudinal process induced stresses more similar to the ones calculated in the 3D model ascompared with the plane strain model.
Keywords Pultrusion process Finite element analysis Residual/internal stress Thermosetting resin
I. Baran () J. H. HattelDepartment of Mechanical Engineering, Technical University of Denmark, DK-2800, Kgs., Lyngby,Denmarke-mail: firstname.lastname@example.org
R. AkkermanFaculty of Engineering Technology, University of Twente, NL-7500AE, Enschede, The Netherlands
C. C. TutumDepartment of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, USA
100 Appl Compos Mater (2015) 22:99118
Degree of cureC0 Critical degree of cure at T = 0 KCT Constant used in the shift factor (f (, T ))C Diffusion constantCpc Specific heat of the compositeCpd Specific heat of the dieE Activation energyEr Instantaneous resin modulusE0r Initial (uncured) resin modulusEr Fully cured resin modulusf (, T ) Shift factor (from kinetics to diffusion region)Htr Total heat of reactionkx1,c, kx2,c, kx3,c Thermal conductivities in the x1-, x2- and x3-directions,
respectively for the compositekx1,d , kx2,d , kx3,d Thermal conductivities in the x1-, x2- and x3-directions,
respectively for the die Constant used in the Di Benedetto equationKo Pre-exponential constantn Order of the cure reaction (kinetic exponent)q Heat source (internal heat generation)c Density of the composited Density of the dier Density of the resinR Universal gas constantRr(, T ) Reaction of curet TimeT Instantaneous temperatureTg Glass transition temperatureTg0 Glass transition temperature of the uncured resinTg Glass transition temperature of the fully cured resinT Difference between the glass transition temperature (Tg)
and the instantaneous temperature (T )TC1 Critical temperature at the onset of the glass transitionTC2 Critical temperature at the completion of the glass tran-
sitionu Pulling speedVf Fiber volume fraction
Pultrusion is a continuous and a cost effective composite manufacturing process in whichconstant cross sectional profiles are produced. While pultrusion machines vary in design,the process is basically the same. Creels of unidirectional (UD) roving provide longitudinaltensile strength in the length of the profile. On the other hand, rows of continuous filamentmat (CFM), woven roving or stitched fabrics provide transverse strength across the widthof the profile. All reinforcements are first fed through the pre-forming guiders which startshaping the fiber reinforcements into the finished product. These reinforcements are then
Appl Compos Mater (2015) 22:99118 101
pulled into a resin bath being wetted out and subsequently entering the heating die. Theheaters initiate the exothermic reaction process in which the resin is being cured. The solid-ified and cured profile is advanced via a pulling system to the cut-off saw where it is cut toits final length. A schematic view of the pultrusion process is given in Fig. 1.
The process induced variations such as residual stresses and distortions are a criticalissue in composite manufacturing , since they affect the structural behaviour and geo-metrical precision of the final product. More specifically, the residual stresses can lead tocracking during curing . In order to capture and investigate these variations, a mechanicalanalysis should be performed.
In literature, thermo-chemical characteristics of the pultrusion process has been inves-tigated numerically and experimentally  in which the temperature and degree ofcure distributions inside the heating die were predicted. All these contributions have onlybeen dealing with thermal modelling in which the temperature of the composite initiallyis lagging behind the heaters temperature; nevertheless during the curing it exceeds thedie temperature due to the internal heat generation of the resin . For this purpose, wellknown numerical methods such as the finite difference method (FDM) and the finiteelement method (FEM)  have been utilized. Using these efficient thermo-chemicalmodels, process optimization studies  as well as reliability analysis  have beenperformed. In , numerical modelling strategies for the thermo-chemical simulation ofthe pultrusion were investigated by the authors where the steady state approach was found tobe computationally faster than the transient approach. In addition to these thermo-chemicalstudies in the literature, state-of-the-art models have recently been proposed by the authorsfor the thermo-chemical-mechanical analysis of the pultrusion  in which the thermo-mechanical aspects including the evolution of the process induced stresses and distortionsin the transverse direction together with the mechanical properties were addressed. In thisnumerical model, a three dimensional (3D) transient thermo-chemical model is sequentiallycoupled with a 2D quasi-static plane strain mechanical model for the pultrusion processof a unidirectional (UD) fiber reinforced profile by using the FEM. The cure hardeninginstantaneous linear elastic (CHILE) approach [20, 21] was utilized for the resin modulusdevelopment. The proposed 3D/2D approach, which was found to be computationally effi-cient, provides an increased understanding of the process by evaluating the development ofthe stresses and distortions as well as the mechanical properties during processing. In ,an integrated modelling of the pultrusion process of a NACA0018 blade profile was carriedout by the authors using the proposed 3D/2D mechanical analysis in . The calculatedresidual stresses were transferred to the subsequent bending simulation of the pultrudedblade profile and the internal stress distribution was evaluated taking the process inducedresidual stresses into account.
3D thermo-chemical-mechanical analysis of the pultrusion process has not been consid-ered in literature up to now. A novel 3D numerical simulation tool embracing the mechanical
Heating Die Puller Saw
Pulling Direction Fibers
Fig. 1 Schematic view of a pultrusion process
102 Appl Compos Mater (2015) 22:99118
aspects of the pultrusion process is hence being developed in the present work. The tem-perature and degree of cure distributions at steady state are first calculated using the 3Dtransient thermo-chemical analysis of a pultruded square product. Afterwards, these pro-files are mapped to the 2D and 3D quasi-static mechanical models. The already developed2D plane strain mechanical model in  is further improved using generalized plane strainelements. Moreover, 3D quadratic brick elements are used for the 3D model for the calcu-lation of the process induced longitudinal stresses as well as transverse stresses. In the 3Dmechanical model, instead of the cross section of the part which is used in the 2D mechani-cal model (see Fig. 3 ), the entire 3D part is assumed to move along the pulling directionof the process while tracking the corresponding temperature and degree of cure profilescalculated in the 3D thermo-chemical simulation (see Fig. 4). Using these three differentmechanical models (i.e. 2D plane strain, 2D generalized plane strain and 3D models), theevolution of the transient stresses and distortions are captured and the obtained results arecompared with each other. The general purpose finite element software package ABAQUS is utilized. The CHILE approach, which is a valid pseudo-viscoelastic approximationof the linear viscoelasticity , is considered for the resin modulus evolution as in .
2 Numerical Implementation
2.1 Thermo-Chemical Model
In the thermo-chemical analysis, the energy equations are solved in 3D space for the UD pul-truded part (Eq. 1) and for the die block (Eq. 2). Here, x3 is the pulling (axial or longitudinal)direction; x1 and x2 are the transverse directions.
t+ u T
+ q (1)
where T is the temperature, t is the time, u is the pulling speed, is the density, Cp isthe specific heat and kx1 , kx2 and kx3 are the thermal conductivities along x1-, x2- andx3-directions, respectively. The subscripts c and d correspond to composite and die, respec-tively. Lumped material properties a