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Home > Library > Finite Element Modeling of 3-D Braided Carbon Fiber/Urethane Elastomer Tubes Finite Element Modeling of 3-D Braided Carbon Fiber/Urethane Elastomer Tubes Christopher Pastore and James Singletary (DuPont) Table of Contents Abstract
This report describes work on the material property and subsequent finite
element (FE) modeling of 3-D braided carbon fiber/elastomeric urethane
tubes, subjected to axial tension and compression, and bending. These
tubes are manufactured by Atlantic Research Corporation (ARC), and are
being evaluated by the Navy for use as flexible moorings. Complexities
of the material include a nonlinear, nearly incompressible matrix, and
high failure strains, which allow significant reorientation of the reinforcing
fibers. Composite structural response is modelled by fitting predictions
from a Fabric Geometry Model (FGM) of 3-D braided tubes for uniaxial and
plane strain loading to the Ogden (generalized Mooney-Rivlin) hyperelastic
law, using a curve-fitting procedure included in Abaqus FE software. The constituents of the tubes are Hercules AS-4 12K carbon tows and Uniroyal Adiprene L-100© elastomeric urethane resin. Table 1 and Table 2 list nominal material properties for the fiber and resin, respectively. Table 1 Fiber Properties of Hercules AS-4 carbon fiber yarn, 12K tows
Table 2 Matrix Properties of Uniroyal ADIPRENE L-100© urethane elastomeric resin
The tubes are braided with Atlantic Research Corporation's 3-D Through-the-Thickness© braiding process, in one pass over a cylindrical mandrel. The result is an integrated tube of carbon yarns. Two candidate fiber architectures have been produced. Table 3 and Table 4 detail Architectures I and II, respectively. Architecture I contains nominally straight, axial yarns in addition to 3-D braided yarns on the bias. Architecture II is similar but contains no axial yarns, and is consequently thinner in braided form. Both architectures have the same nominal braid angle of 40°, and the same nominal total fiber volume fraction of 60%.
The test protocol for tension and compression parallels ASTM standards:
There is no analogous standard for testing in bending or combined tension/bending (see below for details of bending tests). Details of the test procedure important for FE modeling are:
FGM/FE Modeling of Material and Structure This section describes the salient features of the constituent materials and structure of the test samples. An hierarchical description of the tubes is used, first as an equivalent, homogeneous material, and then as a structure composed of this material. The former step -- material homogenization -- used stiffness averaging, outlined in Section 4.1. The structural analysis was then carried on using Abaqus 5.6 FE software, and is outlined in Section 4.2. We believe this combination of analysis techniques is novel, and reduces the complexities of the actual part to a tractable description. These complexities include:
These complexities combine in a nontrivial way. FGM Description of an Equivalent, Homogenized Material We assumed the actual, composite tube may be treated as an equivalent, homogeneous, nonlinear material. This is justified because the dimensions of the unit cell are much smaller than the dimensions of the test specimen, and given the large distance over which experimental strains will be measured, should introduce negligible error (c.f. examinations of error due to finite unit cell size in analyses of angle-interlock woven graphite epoxy composites in Lamattina, of triaxially-braided graphite/epoxy composites in Masters et al., of triaxially-braided glass/urethane composites in Singletary, and theoretical analyses of strain variations due to unit cell size in various textile composites in Bogdanovich and Pastore). Byun and Chou, Chou and Ko, Gowayed and Pastore and Bogdanovich and Pastore review various models to predict the response of such homogenized equivalents of textile composites. The roots of these models can be traced to averaging methods, first proposed by Voigt. Averaging methods bound the equivalent material response with the fact that it can be proven, on variational principles, to lie between the upper bound of the volume-weighted average of the constituents' stiffness tensors, and the lower bound of the volume-weighted average of the constituents' compliance tensors (c.f. Hashin and Rosen). It is seen that, in a textile composite, as the reinforcing fibers approach the load direction in uniaxial load, the upper bound, stiffness averaging, is a good estimate. We chose to use the upper bound of pure stiffness averaging, as originally proposed for textile composites by Kregers and Melbardis. This choice was based on previous experience by the investigators in analyzing tension test of cylindrical braided specimens; changing it should result in more compliant predictions, but would not affect the analysis method. Singletary has found that for cut braided specimens (wherein the yarns at the edges of the test coupons have been cut), a blend of combination of stiffness and compliance averaging seems most appropriate. However the given specimens are in cylindrical coupons which have no cut edges, thus behave more in accord with the assumptions of stiffness averaging. Succinctly, an idealized unit cell for the material is derived by geometric arguments based on processing parameters, such as yarn size, number, and braid angle. The unit cell has a finite number of yarn orientations. Each orientation is assigned a relative volume, based on the length of yarn in that direction relative to the total length of yarn in the ideal unit cell. The transversely isotropic stiffness tensor of an impregnated yarn is calculated from a micromechanics model, using the global fiber volume fraction (we chose the micromechanics model of Vanyin,. It is then transformed to each orientations in the ideal unit cell. These individual stiffness tensors are multiplied by the corresponding relative volume fractions, and then summed to form the stiffness average of the homogenized, global response. Several sources in the literature ( e.g. Kregers and Melbardis, Gowayed and Pastore, Masters et al., Singletary, Bogdanovich and Pastore) give further details and examples of the stiffness averaging method. Because of material and geometric nonlinearity, different stiffness predictions were made at idealized deformation states - differing axial strains in uniaxial and plane strain tension and compression. The geometry of the idealized unit cell at a defined axial strain in uniaxial or plane strain determined the orientations, and thus stiffness tensors, of the idealized, impregnated yarns. High matrix Poisson's ratio and fibrous reinforcement made planar response much stiffer than uniaxial. Matrix stiffness was assumed to be a function only of the axial strain. This was a gross estimate for hyperelastic materials, but no other data for the matrix was available. Further, strains to failure were rather low, suggesting that the error accumulated from this assumption is relatively small. The modelling of the stress-strain response using the FGM approach included recalculation of the braid angle, through thickness angle, fiber volume fraction and percent longitudinals (for triaxial braids) as a function of assumed applied axial strain. This was calculated by considering the geometric deformations from Poisson effects and the associated volumetric deformations of the specimen as a whole. The code was written in Visual Basic and executed using Excel.
The triaxial braid predicted tensile and compressive stress-strain response are compared with experimental data in Figure 3. The predicted compressive and tensile responses are virtually indistinguishable despite the differences in matrix response because the longitudinal yarns overwhelm the stiffness response. The buckling is not predicted in the FGM method, so there is a noticeable discrepancy between the prediction and experiment at the point of buckling.
Figure 3 Nominal uniaxial tension and compression stress-strain curves for triaxial braided composites compared with experimental data. Uniaxial FGM predictions for the biaxial braid are compared to test data obtained by ARC in Figure 4 for tube tension and compression. Owing to its off-axis reinforcement, Architecture II is the more nonlinear material, and has higher failure strains. The experimental stress-strain curves show only the elastic response of the tubes; tests showed significant load-carrying ability after initial failure.
Figure 4 Nominal uniaxial tension and compression stress-strain curves for Architecture II, as predicted by FGM and as measured experimentally.
It should be remembered that experimental data takes strain as crosshead
travel, therefore, the inhomogeneous deformation expected of a clamped,
nearly-incompressible material which necks or bulges significantly is
included in the experimental data, but not in the FGM, which assumes
homogeneous, uniaxial load. There is significant scatter in the data,
and it is therefore difficult to say which, if any, of the curves can
be taken as "typical." However, the FGM model seems to predict
something like an average response in tension. The FGM predicts significant
strain-stiffening, attributable to fiber reorientation. Experiments do
not seem to show as much stiffening. In compression, the FGM predictions
are in good agreement with experiment at low strains, but significantly
over predict stiffness at higher strains. The FGM material description was then used as a basis for a FE structural
analysis. Since the choice of several key variables and factors, such
as element definition, material definition, mesh definition, and nonlinear
analysis procedure, are typically somewhat arbitrary, it is often difficult
to conceive of an ideal FE analysis. In this case, many of these
factors were constrained by convergence considerations and memory space,
and therefore the resulting model is serendipitously closer to an ideal
model than might normally be expected. Model Geometry and Boundary Conditions For bending, half of the cross-section and half of the tube length is
needed. Although compression and tension could be modelled as axisymmetric,
the more general, 3-D bending model is used exclusively. Figure 5 shows
an Abaqus plot of a typical FE model of this half-tube.
Figure 5 Tension mesh for FE modelling Figure 6 illustrates the mesh used for compression and bending in the FE model.
Figure 6 Basic mesh for compression and bending modelling
FE model boundary conditions for compression and bending are shown in Figure 7. Loads are imposed as boundary condition displacements.
Figure 7 Boundary conditions for compression modelling. Bending is similar with different load based conditions on central end. Convergence considerations arose mainly from the near-incompressibility
of the elastomer composite. In order to enforce incompressibility, a hybrid
displacement/pressure formulation interpolation scheme was necessary.
Because of the expected radial displacement, shell elements, whose bending
stiffnesses require arbitrary definition were ruled out in favor of 3-D
hexahedral elements. Experience verified advice given in the Abaqus User's
Manual, that both full integration and second-order
elements were required for a convergent model. These latter two constraints
increase computational time and greatly increase memory required for solution.
Taken together, these considerations forced the use C3D20H hexahedral
elements. As seen in Figure 3 and Figure 4, the composite response resembles
the response of a homogeneous, hyperelastic material, albeit a very stiff
one. The composite is also expected to be very nearly incompressible.
We therefore believe it is reasonable to describe material response as
a homogeneous, hyperelastic material, using classical, phenomenologically-based
hyperelastic laws. Abaqus allows the use of the Polynomial and the Ogden
(or generalized Mooney-Rivlin) law, from which we found the more flexible
Ogden law to give a better fit to the FGM data. Data points in the form
of nominal stress versus nominal strain were input into the model directly,
and were fitted by a nonlinear least squares algorithm internal to Abaqus.
It was found that both uniaxial and planar FGM predictions, in tension
and compression, were necessary in order to produce a stable set of hyperelastic
constants. It was possible to include higher-order terms in order to better
capture inflection points in the stress-strain curve. Trial-and-error
showed the use of higher-order terms negligibly affected the FE predictions,
but did adversely affect stability, and so the lowest order of the fit
was used. The small effect of order of fit on FE predictions is understandable,
given the smoothness of the FGM data. Details of the Ogden law and the
input data curve-fitting procedure are given the Abaqus manual. The mesh density was constrained by memory space available, which was
about 200 MB. The expensive elements and expensive, nonlinear material
description therefore limited mesh refinement. With such elements, the
model was limited to about 3000 DOF. The final model selected had 2238
displacement and 768 pressure DOF, with mesh refinement near the clamped
end. The model was loaded by displacement boundary conditions and analyzed
at displacement from no-load equilibrium at regular intervals, up to 10%
strain in tension and compression. Analysis was Newton-Raphson implicit
integration. Comparisons between experimentally determined values and FE/FGM predictions are shown below. The model seems to predict low-strain response in both compression and tension well. The model predicts reasonable stresses at higher tensile strains, but drastically overpredicts stress at high compressive strains, as did the FGM predictions alone. Figure 8 shows uniaxial tension for both biaxial and triaxial specimens.
The noted deviation in the experimental value of biaxial tension is due
to the presence of an internal restraint which was not modelled in the
finite element.
Figure 8 Comparison of Experimental and FGM/FE predictions of uniaxial tension
Figure 9 shows a comparison of experimental and FE/FGM predictions for uniaxial compression. As can be seen, the triaxial specimen shows buckling at relatively low strains. The model formulation would track buckling if it were predicted to occur, however, the FE model appears too stiff to permit significant buckling.
Figure 9 Comparison of experimental and FGM/FE predictions of uniaxial compression
However, plots of the deformed shape of the FE mesh at higher compressive
strains show ripples forming near the clamped end, as indicated in Figure
10.
Figure 10 Axial stress distributions in triaxial specimen during axial compression
Figure 11 indicates the comparison between experimental and FE/FGM predicted bending response. The transverse displacement and transverse load are plotted in this figure. Although this response is rather linear, it should be noted that the FE/FGM predictions of axial stress during this deformation process are markedly non-linear, as would be expected for the biaxial material.
Figure 11 Comparison of experimental and FGM/FE predictions of bending. The model was found to be most sensitive to the type of elements used. The near-incompressibility of the matrix required a hybrid displacement/pressure formulation. Experience showed that using linear ( i.e. 8-noded) elements lead to hourglassing, which necessitated the use of quadratic, 20-noded bricks. Furthermore, reduced-integration hexahedral elements lead to divergence at the clamped end in compression, requiring the use of full integration elements, as suggested in the Abaqus manual. The combined effect of using higher-order and fully-integrated elements was to increase runtime and greatly increase scratch memory space required. The latter proved a difficulty for this project, as locker space of about 200 MB limited the size of the model which could be examined, hampering convergence studies. Because of the incompleteness of convergence studies, FGM/FE predictions should be taken with some degree of caution. On a philosophical level, however, it is not often that problems of academic interest can be modelled with a small enough model to allow resources for a detailed convergence study. We believe our FGM/FE method to be an appropriate compromise between rigor and engineering approximation. Using FGM predictions allows a dependency in the material properties directly on the controllable processing conditions, such as braid angle and wall thickness. The FGM is simple enough to run on a spreadsheet, so the cumbersomeness of having a preprocessor for material properties is somewhat offset. Fitting the FGM data to the FE hyperelastic model appears reasonable given: 1) the incompressibility of the matrix, and 2) the goodness of the fit between hyperelastic laws and FGM predictions for simple deformations in a single-element FE model. Using FE allows complicated structures to be analyzed quickly, and can
show how structural response interacts with the local, material response
in a way not possible with FGM or some other homogenization technique
alone. This is demonstrated in the compression of the cylinders (Figure
9). FGM agrees with the experimental, structural response (as measured
by load cell and crosshead travel) at low strains, but substantially overpredicts
stiffness at higher strains. The FE model demonstrates that the material
is actually beginning to buckle, and the additional compliance thus may
be a structural and not a material phenomenon. At this time, we have been
unable to view either actual tests or videotapes made of previous tapes,
to confirm whether or not the actual test specimens buckle. Criticisms of the FGM/FE Model Two criticisms of the FGM/FE model are the use of stiffness averaging, and the incorrectness of fitting an anisotropic material response to a deformationally-isotropic hyperelastic law. The choice of stiffness averaging was arbitrary. Stiffness averaging predicts an upper bound to material stiffness, and thus in general overpredicts stiffness. For textile composites whose primary reinforcement is oriented with the load, the agreement between stiffness averaging and measured response has been seen to be quite good ( c.f. Masters et al. or Singletary). However, as the relative volume fraction of fibers off-axis to the load increases, the composite response drops from the stiffness-averaging upper bound. Several authors have suggested empirical combinations of upper- and lower-bound estimates for the response of equivalent materials with significant off-axis reinforcement, as is the case with Architecture II (see e.g. Pochiraju et al., Singletary, and Bogdanovich and Pastore). We justify our choice of pure stiffness averaging by the good agreement it gives with the limited experimental data. Our methodology would not be changed substantially if a different averaging method were chosen. Hyperelastic laws assume that the undeformed material is isotropic, which
is clearly not true of composites. Our fitting of anisotropic FGM data
to a hyperelastic law violates the assumption of initial isotropy. We
justify this because the data desired from the model, namely crosshead
travel and resulting forces, are relatively insensitive to composite anisotropy.
In the course of this investigation, several other methods were examined and then abandoned for various reasons. The first was the addition of reinforcement into the elastomer matrix directly via the *REBAR option, which allows elements to be reinforced by straight rods of arbitrary stiffness at arbitrary volume fractions and directions. This method is suggested for reinforced concrete, and is stated to be acceptable for reinforced rubber. However, experience with the *REBAR option showed it is applicable only for dilute reinforcement in the loading direction, and not general use. There are to reasons for this. First, the model assumes that the longitudinal stiffness of a unidirectionally-reinforced element is El = Vf Ef + Em, which neglects the fact that including reinforcement must exclude matrix. Second, the model assumes that reinforcement does not affect transverse stiffness, so that in the limit, Et = Em. This gives rise to drastic under predictions of the reinforcement value of off-axis reinforcement like the 40° braiding yarns. With these limitations, the *REBAR option was judged to not be useful. We considered fitting experimental data directly to the hyperelastic laws, using the automated curve-fitting scheme internal to Abaqus. This idea seemed to offer an inappropriately low level of erudition. It allowed no direct dependence of model predictions on processing parameters. Finally, given the scatter of the data, it is not clear which curves should be taken as typical of composite response. A third option was to use the Ramberg-Osgood deformational plasticity law, assumed rather arbitrarily in Potty et al.. The Navy had objections to this option, and we agreed that it is inappropriate to estimate non-proportional loading as in compression and bending by a deformational plasticity theory which assumes proportional loading. A fourth option was to write a user-defined material law subroutine. However, this required being able to write a function for the Jacobian of the stiffness tensor at any, arbitrary deformation state. Our FGM basis only allows the predictions of the Jacobian under idealized deformations, and we could not easily conceive of how to generalize the FGM to a function of arbitrary deformations for a large-deformation problem. A fifth option pursued and ultimately abandoned was to assume the carbon fibers are rigid restraints on the compliant, hyperelastic matrix. Given the ratio Ef,l/Em, this is a reasonable assumption for Architecture II (with no axial yarns). A mesh with elements in the shape of the unit cells was defined, and multipoint constrain equations, using both the *MPC and the *RIGID LINK options, were defined to impose the rigid restraint of the braider yarns on the element corners. This option lead to divergence problems, presumably because the additional constraint equations over constrained the system. Over constraint of nearly-incompressible materials appears to be a common problem with hyperelastic FE analyses (e.g. Cook et al.). A sixth option was to use shell elements, composed of layers of hyperelastic material and of orthotropic, elastic material which represented fibrous reinforcement. This approach has been used to model reinforced rubber which does not undergo much fiber reorientation (Furdson) but cannot capture the geometric nonlinearity of fiber reorientation. As is typical of composites analyses, our model depends on the adequate description of the constituent fiber and matrix for extrapolation to an equivalent, homogeneous material via FGM, and from this to a structural response via FE. AS-4 carbon fibers are common materials, and their elastic constants can be found in the literature (see e.g. Kawabata, or Bogdanovich and Pastore). Adiprene L-100© elastomer, however, is not described in either the common literature or in Uniroyal publications on elastomers. As an elastomer, it is expected to be nonlinear, with different response in tension and compression. Elastomeric behavior is very dependent on chemistry and cure cycle (see the Abaqus manual). The behavior of similar elastomers, for which more data exists, cannot be expected to give accurate insight into the matrix used here (see Palinkas). An accurate fit of this nonlinear, unsymmetric stress-strain response to hyperelastic models requires stress-strain curves from multiple deformation states (see the Abaqus manual). We have therefore asked for uniaxial, planar, or biaxial tensile and compressive test data. All that was received was compressive uniaxial response, on which the predictions are based. Experimental data for composite tubes tested at ARC shows significant variation, and their verification program calls for only a handful of experiments for any architecture in any loading. These factors combine to give a very low confidence level for any values assumed to be `average'. Therefore we have not performed any statistical analysis of the goodness of the fit between our predicted results and experimental data. Finally, it should be remembered that the elastic limits were inferred somewhat arbitrarily from the experimental stress-strain curves supplied, not determined more directly from other means such as cyclic testing or acoustic data. A failure mode of gradual, progressive debonding of yarns from resin pockets, until debonding cracks coalesce into a catastrophically large flaw is consistent with the stress-softening and then sudden failure seen in the experimental data at high strains in both tension and compression, and could explain some of the difference between our predictions and experimental data. A Fabric Geometry Model was coupled to a Finite Element model to predict the structural response of elastomeric urethane tubes reinforced with 3-D braided carbon fibers. The observed material response was complicated by near-incompressibility and large-strain considerations which included material nonlinearity in the matrix and geometric nonlinearity in the reinforcement structure. Predictions of the combined FGM/FE approach agree with the limited set of experimental data available. References K. Potty and J. W. Gillespie and T. A. Bogetti, Final Report on Modeling MOB Compliant Coupling Response, Center for Composite Materials, University of Delaware, Sep. 18, 1996 A. E. Bogdanovich and C. M. Pastore, Mechanics of Textile and Laminated Composites with Applications to Structural Analysis, Chapman Hall, 1996, London B. Lamattina, Preforming, RTM processing and textile characterization of 3-D angle interlock carbon/epoxy composites, MS Thesis, University of Delaware, 1993 J. E. Masters and Y. A. Gowayed and C. M. Pastore and R. L. Foye, "Mechanical properties of triaxially braided composites: Experimental and theoretical results", Journal of Composites Technology and Research, 15, 2, pages 112-122, 1993. James Singletary, Characterization of the Elastic Properties of Triaxially Braided E-Glass/Urethane Composites, MS Thesis, North Carolina State University, Raleigh, NC, 1994. J. H. Byun and T.-W. Chou, "Modelling and Characterization of Textile Structural Composites: A Review", Journal of Strain Analysis, 4, July-August, 1989. T.-W. Chou and F. K. Ko, Textile Structural Composites, Elsevier, Amsterdam, 1989. Yasser. A. Gowayed and Chris. M. Pastore, "Analytical Techniques for Textile Structural Composites: A Comparative Study of US-USSR Research", Proceedings of FIBER-TEX 90, NASA, CP 3128, 1990. Woldemar Voigt, Lehrbuch der Kristallphysik, B. G. eubner, Leipzig, Germany, 1928. (in German) Zvi Hashin and B. Walter Rosen, The Elastic Moduli of Fiber-Reinforced Materials, Journal of Applied Mechanics, June, pp 223-232, 1964. A. F. Kregers and Yu. G. Melbardis, "Determination of the Deformability of Three-Dimensionally Reinforced Composites by the Stiffness Averaging Method", Polymer Mechanics, 14, 1, Jan-Feb, pp 3-8, 1978. G. A. Vanin, Micromechanics of Composite Materials, Durya, Kiev, 1985 (in Russian). G. A. Van Fo Fy, "Elastic Constants and State of Stress of Glass-Reinforced Strip", Polymer Mechanics, 2, 4, Jul-Aug, pp 593-602, 1966. ABAQUS User's Manual, Hibbitt, Karlsson, and Sorensen, Inc., Version 5.6 K. Pochiraju and A. Parvizi-Majidi and T.-W. Chou, Process-Microstructure-Performance Relationships of 3-D Braided and Woven Textile Structural Composites, Quarterly Report, NASA Advanced Composites Technology Mechanics of Textile Composites Work Group, Langley, {VA}, April-June, 1993. R. D. Cook and D. S. Malkus and M. E. Plesha", Concepts and Applications of Finite Element Analysis, John Wiley and Sons, Inc., 1989. P. M. T. Fursdon, "Modelling a Cord Reinforced Component with ABAQUS", 6th UK ABAQUS User Group Conference Proceedings, 1990. S. Kawabata, "Measurement of the Transverse Mechanical Properties of High Performance Fibres", Journal of the Textile Institute, 81, 4, pp 433-447, 1990. R. Palinkas, Materials characterization in the design of castable polyurethane parts, Uniroyal Chemicals Company, Inc., Middlebury, CT 06749
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