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Axial crush simulation of braided carbon tubes using MAT58 in LS-DYNA
ARTICLE IN PRESS Thin-Walled Structures 47 (2009) 740–749
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Axial crush simulation of braided carbon tubes using MAT58 in LS-DYNA Xinran Xiao , Mark E. Botkin, Nancy L. Johnson General Motors Corporation, R&D Center, MC 480-106-224, 30500 Mound Road, Warren, MI 48090-9055, USA
a r t i c l e in f o
a b s t r a c t
Article history: Received 17 October 2007 Accepted 14 December 2008 Available online 4 February 2009
This paper examines a composite damage constitutive model, MAT58, in LS-DYNA and its application for use in braided composite tube axial crush simulations. The constitutive response of MAT58 was investigated using single element simulations. It was found that MAT58 reproduced the softening behavior of the braided composite under monotonic compressive loading, but failed in subsequent unloading and tensile loading cycles. A deficiency in the damage law in MAT58 was identified. Unloading and reloading a volume of material that had suffered some degree of damage was a part of the process with the progressing of crush zone during the axial crush of composite tubes. Consequently, this deficiency hinders the success of MAT58 in such applications. In tri-axial braided composite tube axial crush simulations, although the predicted initial peak forces were within 20% of the experimental values, the predictions for the specific energy absorption (SEA) values were consistently low, particularly for tubes without a plug as crush initiator. These discrepancies are attributable to the deficiency in the damage law in MAT58. & 2008 Published by Elsevier Ltd.
Keyword: Carbon fiber composites Dynamic finite element simulation Damage Crashworthiness Tube crush Energy absorption
1. Introduction The exceptionally high specific energy absorption (SEA) of fiber reinforced polymer composites [1] had long captured the imagination of automotive design engineers at Detroit [2–4]. Even though composites are common in primary structural components in racing cars, the challenges in developing automotive production-feasible composite vehicles are formidable. Towards this common goal, the three Detroit-based automakers, Ford, Chrysler and General Motors, formed the Automotive Composite Consortium (ACC) in 1988. One of the challenges in structural requirements for a passenger vehicle is the crashworthiness requirement. ACC has demonstrated in its first focal project (FP-I) that composite front structures can meet the crash safety requirements [5,6]. In FP-I design, the crash response and crash force of the front rails were determined by using a lumped-mass model with vehicle dynamic analysis [6]. The information was then fed into a finite element collapsible beam code to determine the load and section properties. Static finite element analysis was employed to determine the stress field. Even though crash simulations using dynamic finite element analysis (FEA) had become routine work in the automotive industry during the course of FP-I program, its applications to composite structures were not successful. Since then, the ACC has been working on developing predictive tools for crash simulation of composite structures [7,8]. In crash
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E-mail address: [email protected] (X. Xiao). 0263-8231/$ - see front matter & 2008 Published by Elsevier Ltd. doi:10.1016/j.tws.2008.12.004
energy management, one critical component of the front structure is the front rail. Extensive experimental studies have shown that braided composite tubes are one of the most promising geometries for the front rail structures [9–15]. Axial crush simulation of braided composite tubes has become a benchmark for the predictive capability of composite crash simulation [7,8,16–18]. It turns out that the crush simulation of composite tubes is a rather difficult task and the progress in the area has been slow. Actually, even under quasi-static loading, failure prediction in laminated composites remains a challenging task. To gauge the predictive capability of laminate composite failure theories, a ‘‘World Wide Failure Prediction of Composite Laminate Exercise’’ (so-called ‘‘Failure Olympics’’) was organized, which involved more than a dozen experts and lasted over 10 years [19]. The final results showed that no one theory could predict all 14 cases given for the exercise although their success rates varied a great deal [20]. The prediction of the composite tube crush is certainly much more challenging since one is dealing with the initial failure as well as the post-peak behavior under dynamic loading and in more complicated material forms. Unlike design analysis where a maximum load, a limiting deformation value, or a structural change satisfying a set of load conditions is the ultimate goal, the vehicle crashworthiness design is aimed at maximizing the energy absorption of certain components while minimizing deformation in the passenger compartment [21,22]. This is achieved by controlled progressive deformation and fracture, load sharing and load transfer between components during deformation. In other words, crash simulation of composites requires the modeling of post-peak response of the materials.
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Post-peak analysis is the key in composite tube crush simulations. After reaching a peak stress at the maximum strength, fabric composites such as braids can undergo large deformations at lower stresses before completely losing the load carrying capability. Local damages certainly have significant impact on the load carrying capacity along the principal stress direction while the load carrying capacity in other directions may or may not suffer. Tube crush also invokes multi-axis stresses in materials. In addition, the strain rate effect of the material may not be negligible. To simulate composite tube crush, one needs constitutive models that can represent these behaviors reasonably well up to the complete fracture of a material. At the same time, one must be able to handle the numerical problems associated with large deformation and fracture. These include element distortion, contact, hourglassing and instability. It is not uncommon for a composite tube crush analysis to abort before completion due to these numerical difficulties. Over the years the ACC has worked with national labs, university researchers and software companies to develop computational tools for the crashworthiness prediction of composite tube structures. The earliest development traced back to 1990 when the ACC sponsored Livermore Software Technology Corporation (LSTC) to implement a composite damage model, MAT58a, in explicit LS-DYNA [7]. The existing constitutive models and the models under development for composite crash application may be classified into two major categories: micro-mechanics models and phenomenological models. A true micro-mechanics model captures the deformation, damage and fracture of individual fiber yarns, matrix, their interface and interaction in a unit cell based on the constituent properties. The model predicts the overall stress–strain response of the unit cell as damage progresses, which is then used in the global model for the element. In reality it is difficult to model every type of damage and its subsequent softening effects in a complicated 3D microstructure. The problem has been approached by adopting simplifications and assumptions at the unit cell level [23–25] or by detailed large-scale modeling [26–28]. Beard and Chang [23,24] developed a micro-mechanics-based model. It calculates the constitutive behavior of a unit cell through an idealized three-dimensional description of the braid geometry using fiber tow and resin properties. The model includes the effect of tow scissoring in braids and considers the damage and fracture in constituents using the maximum stress criterion. It updates the effective stiffness of the unit cell as damage progresses. The model has been implemented in implicit [23] and explicit [25] ABAQUS and used for analysis of tube crush with a plug initiator. Quek et al. [26,27] and Song et al. [28] developed a detailed finite element (FE) model for a representative unit cell (RUC) in a braided composite. The RUC predicted softening responses for tri-axial braided composites under compression. Introducing imperfection into the RUC further improved correlations between simulations and experimental results. Phenomenological models, on the other hand, describe the global constitutive behavior. In fact, most of the material models in FE codes are phenomenological models. Commonly used piecewise linear plasticity model for metallic materials is a good example. Instead of modeling plastic deformation resulted from a complicated set of deformation mechanisms such as the motion of individual dislocations, their interactions with barriers such as grain boundaries and precipitations, and interaction between dislocations, etc., the plasticity model describes the nonlinear deformation through a stress–strain relation measured at the macroscopic level. The continuum damage mechanics-based phenomenological models are characterized by the use of damage variables and
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damage evolution laws. The damage variables are internal variables that cannot be measured readily by standard material tests. The code developers often provide insufficient information about how to obtain these parameters. Consequently the parameters in damage models in FE codes are determined by curve fitting, so-called calibration. This is further complicated by the fact that the standard composite material tests generally do not provide information beyond the peak load. Even though some composite structures, such as tubes made of fabrics, exhibit significant load carrying capacity after initial peak load, such behaviors were not recorded in material tests. To model the crash responses of these composite structures, the material must have some post-peak load carrying capabilities. Therefore, various forms of post-peak behaviors were assumed. The damage parameters were selected by best fit to a component test curve. The problem is that a set of parameters obtained under one condition often does not work for the next condition. This prohibits the predictive capability of the analysis. Physics-based composite damage models have been developed for laminated composites and implemented in commercial dynamic FE codes. The CODAM model developed by Williams et al. [29,30] is an example. An oversized CT specimen was used to generate specimens with different levels of damage to obtain the parameters in CODAM needed for post-peak response under tensile load [31]. A proper material testing method is the key in developing physics-based phenomenological models. To authors’ best knowledge, the work by DeTeresa et al. [32] at the Lawrence Livermore National Lab (LLNL) was the first to generate damage data using un-notched specimens as in a material test on fabric composites. Using LLNL data in MAT58 of LS-DYNA, we obtained encouraging results in recent attempts in tube crush simulations. The results of this new initiative are summarized in this paper.
2. Composite material model—MAT58 MAT58 in LS-DYNA [33] was based on Matzenmiller’s damage mechanics model [34]. The model has the capability of modeling the damage independently in the principle directions of orthotropic materials. For the plane stress condition, the constitutive equation is given as [7]: 2 3 3 2 3 2 s1 1 ð1 o1 Þð1 o2 Þn21 E2 0 ð1 o1 ÞE1 6 7 16 7 6 7 6 s2 7 ¼ 6 ð1 o1 Þð1 o2 Þn12 E1 7 7 6 ð1 o ÞE 0 2 2 5 4 2 5 4 4 5 c
0
t
0
c ¼ 1 ð1 o1 Þð1 o2 Þn12 n21
cð1 os ÞG
g
(1)
In Eq. (1), s is the normal stress, t is the shear stress, e is the normal strain, g is the shear strain, E is the modulus, G is the shear modulus, n is Poisson’s ratio, and o is the damage function. As a special case, an exponential damage evolution law was provided mi
oi ¼ 1 eði =0i Þ
(2)
with i ¼ 1T, 1C, 2T, 2C, S (T ¼ tensile, C ¼ compressive, S ¼ shear). In Eq. (2), mi and e0i, are sets of parameters. e0i is related to other parameters and properties by
0i ¼
X t;c ðmi eÞ1=mi Ei
(3)
where X is the strength. The parameter mi controls the shape of the stress–strain response. Fig. 1 shows the stress–strain curves for different mi values. Eq. (3) was implemented in MAT58 of LS-DYNA in its earlier version [23], MAT58a.
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1.2
where a is the ratio of the limiting stress to the peak stress, 0oap1.0. a ¼ 1.0 corresponds to no strain softening. Eq. (4) gives the user an option to avoid the localization. Typical stress–strain curves for various values of Ei and SLIMi are shown in Fig. 2.
1 m = 0.25 m = 0.5 m=1
0.6
3. Lawrence Livermore National Lab (LLNL) test data
m=2 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
Strain Fig. 1. Examples of stress–strain responses for various m values, MAT58a.
1.2 1.0
Stress
0.8 0.6 0.4 E11t = 0.01, slimt1 = 0.5 E11t = 0.02, Slimt1 = 0.5
0.2
E11t = 0.03, Slimt1 = 0.7 Elastic
0.0 0
0.02
0.04
0.06
0.08
0.1
Strain Fig. 2. Examples of stress–strain responses for various Ei and SLIMi values, MAT58b.
One of the shortcomings of MAT58a is that not only the postpeak response but also the pre-peak response varies with the mi value. As shown in Fig. 1, to simulate a material with relatively large post-peak deformation, one needs to select a smaller mi value to obtain a slower unloading curve. A smaller mi, however, results in a softer pre-peak response, which deviates significantly from the nearly elastic behavior exhibited by common continuous fiber reinforced composites. The direct coupling between the preand post-peak stress–strain relation compels the analyst to sacrifice one response to satisfy the other. Later, the damage evolution law in MAT58 was modified [35]. The modified version is MAT58b. In MAT58b mi is not employed. Instead, two new sets of parameters Ei (i ¼ 11T, 11C, 22T, 22C, S) and SLIMi (i ¼ T1, C1, T2, C2, S) are introduced. Ei is defined as the strain at the maximum stress response and SLIMi as the minimum stress limit of damaged material. A parameter study indicated that varying Ei changed the slope of the pre- and post-peak response, the same way as mi in MAT58a except that Ei appears to be reciprocal to mi, i.e. a greater Ei value results in a smaller slope of the stress–strain response. SLIMi sets a predefined limiting stress. At the limiting stress, the damage law is described by [35]
o¼1
aX t;c E
(4)
To support the constitutive model development for composite tube crush applications, DeTeresa et al. [32] measured the socalled ‘‘total stress–strain responses’’ of a braided carbon composite under tensile and compressive loading. The flat plaques used in the study were 0/730 tri-axial braid manufactured using resin transfer molding. The braid contains 80 k carbon fiber tows in both axial and bias direction. The matrix is an epoxy vinyl ester resin. Both tensile and compression tests were conducted using straight-sided specimens. The tensile test was similar to a standard test except the specimens were pulled until complete fracture (zero load) was attained. For the compression test, rather than using a standard tabbed test specimen and a compression fixture for fiber composites, untabbed specimens were gripped directly with hydraulic grips in a servo hydraulic test machine. With a short gage length, acceptable compressive failure was achieved. An apparent strain was calculated from the machine displacement and the specimen gage length. The specimens were compressed well beyond the initial peak load until about 15–20% strain. Fig. 3 plots a set of compressive stress–strain curves measured in [32]. These data reveal that the braided composite had retained a significant fraction of its load carrying capacity after having reached the commonly defined failure strength, particularly under compressive loading. The post-peak responses of this composite resemble those depicted by MAT58b in Fig. 2, suggesting that the parameter SLIMi may be estimated from the total stress–strain curves. DeTeresa et al. [32] also measured the total stress–strain curves for off-axis angles and 901 specimens at an intermediate strain rate of 1/s. They reported that the strengths were about 30% higher than those measured in quasi-static tests. The strains at maximum stress were also higher when compared with the quasistatic values. The modulus was not measured in the strain rate test. Table 1 provides the mechanical properties of the tri-axial braided composites. Table 2 lists the subset of data relevant to that needed for MAT58 extracted from the results of [32]. It shows
16 14 Compression Stress (ksi)
Stress
0.8
12 10 8 6 4 2 0
0
0.05 0.1 Apparent Compression Strain (in./in.)
0.15
Fig. 3. Compressive stress–strain response of triaxial braided panel specimens in 01 direction, 4 repetitions [32].
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Table 1 Mechanical properties of triaxial braid 0/+30/30, Fortafil. Mechanical properties
Ultimate strength (MPa)
(Chord) Modulus (GPa)
Strain to failure
Poisson’s ratio
01 tensile 901 tensile 01 compression 901 compression
305.4 32.1 120.7 66.3
42.3 5.7 32.2 8.6
0.0104 0.0097 0.0039 0.0085
1.169 0.207
Table 2 Data extracted from DeTeresa et al. [32] for MAT58 application. Type of test
Strain rate
Maximum stress (MPa)
Ei, strain at maximum stress
SLIMi, ratio of min. stress/max. stress
01 tensile 901 tensile 901 tensile 01 compression 901 compression 901 compression
Quasi-static Quasi-static 1/s Quasi-static Quasi-static 1/s
299.2 24.1 38.0 93.7 56.9 71.7
0.01 0.009 0.02 0.01 0.014 0.025
0.05 0.57 0.43 0.44 0.42 0.34
60 40 20
Stress (MPa)
0 -20 -40 -60 -80 -100 LLNL test MAT58
-120 -140 -0.06
-0.04
-0.02
0
0.02 Strain
0.04
0.06
0.08
Fig. 4. Comparison of the MAT58 simulation to LLNL compression–tension test data.
that for the braided composite tested at LLNL, the SLIMi values are between 0.3 and 0.5. The 01 tensile test is an exception, which yielded a low SLIMi value about 0.05.
4. Single element test In the LLNL report, DeTeresa et al. [32] presented the results of a compression–tension test. The specimen was loaded first in compression beyond the onset of failure and then reversed to tension. The recorded stress–strain lotus is plotted in Fig. 4. DeTeresa et al. noted that the compressively damaged material did not sustain a tensile load until most of the compressive displacement was recovered. A single element model was used to examine whether MAT58 could reproduce the stress–strain lotus of the LLNL test. In the simulation, the element was compressively loaded through prescribed nodal displacements to the same maximum strain level as in the test and then the direction of the displacement was
Fig. 5. Tube crush experiment setup.
reversed. The element stress–strain lotus is compared with the test data in Fig. 4. As seen, the stress–strain response obtained by simulation matched the test curve reasonably well over the compressive loading segment but poorly over the unloading and tensile loading segments of the curve. Matzenmiller’s damage mechanics model [34] treats the softening behavior through modulus reduction. This treatment is computationally efficient, easy for implementation in FE codes and is also sufficient to represent softening behaviors of damaged materials under monotonic loading. To predict the energy absorption, however, one needs to model not only the loading but also unloading response of damaged materials. The elastic modulus in unloading segment affects the elastic energy released by the material. Because the energy conservation is sustained in dynamic analysis, an over-predicted elastic energy release changes the energy
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Table 3 Notation for tube configurations. Notation
Tube configuration
F[0/730] F[0/730]2 F[0/730]4 F/G[0/730]2 F/G[0/745]2 F/G[0/760]2
1-ply 2-ply 4-ply 2-ply 2-ply 2-ply
triaxial triaxial triaxial triaxial triaxial triaxial
braid, braid, braid, braid, braid, braid,
0/+30/30, 0/+30/30, 0/+30/30, 0/+30/30, 0/+45/45, 0/+60/60,
Fortafil Fortafil Fortafil Fortafil Fortafil Fortafil
80 k 80 k 80 k 80 k 80 k 80 k
tows tows tows axial tow, Grafil 12 k bias tows axial tow, Grafil 12 k bias tows axial tow, Grafil 12 k bias tows
distribution of the system. The released elastic energy will be fed back into the system causing a sudden increase in kinetic energy and hence a greater tendency of instability of the system.
5. Tube crush test The axial crush experiments of the composite tubes were conducted using a drop tower [36]. Fig. 5 shows a tube crush experiment setup. The tube was mounted to the drop head and raised to the predetermined height and then the drop head was released to fall under gravity. The tube was crushed against a steel plate. The force was calculated from the recording of four load cells positioned under the steel plate. The velocity and displacement were calculated from the signal of an accelerometer attached to the drop head. The data acquisition system had a sampling rate of 20 kHz, i.e. an interval of 50 ms. For the crush experiments reported in this paper, the total drop mass was 140 kg and the drop height was 2.54 m. The recorded initial velocity was 7.1 m/s. The composite tubes were 360 mm in length. To ensure progressive failure, the crush end of the tube was chamfered with a 451 bevel. The tubes considered in this study had a square cross section of 55 55 mm2. Two sets of composite tubes were considered in this work. The first set was 0/+30/30 tri-axial braided 50 50 mm2 square tubes. Fortafil 80 k carbon fiber tows1 were used as axial and bias tows. The resin was epoxy vinyl ester resin. The tubes were made of 1-ply, 2-ply and 4-ply of braids. These tubes will be identified as F[0/730], F[0/730]2 and F[0/730]4 hereafter in the text. Table 3 gives a summary of the notation for the tube configurations used in this study. Composite tubes were manufactured using the resin transfer molding (RTM) process [10]. The average thickness of the molded composites varied between 2.0 and 2.4 mm per ply. Flat panels of the tri-axial braided composite were also manufactured. The mechanical properties of the composite were tested using specimens cut from flat panels. The second set of tubes included 2-ply 50 50 mm2 tubes of Fortafil/Grafil tri-axial braid of 0/+30/30, 0/+45/45 and 0/+60/ 60 configurations, i.e. F/G[0/730]2, F/G[0/745]2 and F/G[0/ 760]2. In these tubes Fortafil 80 k fiber tows were used as the axial tows and Grafil 12 k fiber tows were used as bias tows. The tubes were manufactured using RTM process. The smaller bias tows resulted in a thinner tube wall. Fortafil tubes yielded an average wall thickness of 2–2.4 mm per ply. Fortafil/Grafil tubes had about 1.1–1.2 mm per ply. Some tubes were tested with a metal plug-type initiator inserted at the beveled end of the tube. Fig. 6 shows a tri-axial braided carbon tube after crush testing and the initiator used for 55 55 mm2 square tubes. The initiator had a 6 mm active radius. 1
80 k tow ¼ 80,000 fiber filaments in a tow.
Fig. 6. Tri-axial carbon tube after crush test.
The plug was attached to the tube using hot glue. In addition, a rubber pad was attached to the bottom of the plug.
6. Tube crush simulation 6.1. Finite element models The FE model consists of three portions: the drop weight, the composite tube and a plug-type initiator for tests with a plug initiator or a rigid wall for tests without a plug initiator. Fig. 7 shows the FE model for a tube with a plug initiator. The drop weight was modeled as rigid with an equivalent weight. The composite tube was modeled using shell elements. The tube was joined to the drop weight through constrained nodes. An initial velocity of 7.1 m/s was assigned to the drop weight and the tube. For tube crush simulations without a plug initiator, the rigid wall force was monitored to compare with the experimental result. The displacement in simulation was taken from the nodal displacement at the center of the upper surface of the drop weight. For tube crush simulations with a plug initiator, the Z-component of the interface force between the tube and the plug was monitored to compare with the force measured in the experiment. The plug-type initiator was modeled as rigid and its lower surface was fully constrained. The multi-layered composite tubes were modeled with multi layers of shell elements and the shell layers were connected through tiebreak contact definition. Details about the tiebreak definition will be discussed later.
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The bevel was modeled with a single row of shell elements per layer. The elements were assigned a tapered thickness such that the two nodes at the leading edge had a smaller shell thickness of 0.1 mm. In the later simulations the bevel was modeled with two rows of shell elements. This change led to a more gradual rise of the initial force response. Three element formulations, Belytschko–Tsay, Belytschko– Wong–Chiang and fully integrated shell [37], were investigated. The more computationally expensive elements did not produce notable improvement to the results and hence, the default element formulation, Belytschko–Tsay, was employed for the analysis. Tube crush simulations were tested with different numbers of through thickness integration points. It was observed that increasing the number of integration points from two to three slightly improved the smoothness of the simulation but added to the computation time. Increasing the number of integration points beyond three did not lead to further improvement. In the end, two integration points were employed in simulations.
6.2. Delamination modeling Delamination was one of the major damage mechanisms in axial crush tests of multi-layered composite tubes. To allow delamination failure, the tube was modeled with multi-layer shell elements connected through the tiebreak contact definition of LSDYNA [33]. Tiebreak contact is a special type of contact. It works the same as common contact types under compressive load. Under tensile and shear loads, tiebreak allows the separation of the tied surfaces following an interface strength-based failure criterion. The
745
following criterion was used in this work:
jsn j NFLS
2
þ
jss j SFLS
2 X1
(5)
where sn is the normal stress, ss is the shear stress, NFLS is the normal failure stress and SFLS is the shear failure stress. The user can choose a failure criterion in other forms including defining a damage curve that allows the bondline deformation before complete separation of the two surfaces. The NFLS and SFLS values cannot be measured directly. In this work these values were estimated through correlating simulations with interlaminar fracture toughness experiments. The NFLS value was obtained by correlating simulations with the load–displacement traces measured in Mode-I fracture toughness experiment using the double cantilever beam (DCB) [38], similar to the method used by Warrior et al. [39]. Fig. 8 shows the FE model for the DCB specimen. It consists of two layers of shell elements connected through tiebreak contact at the base section. Adjusting the NFLS value, we can match the peak values of the load–displacement curves obtained by simulations with the experimental data. Similarly, the SFLS value was obtained through correlating simulations with the end notch flexural (ENF) experimental results that measures the interlaminar Mode-II fracture toughness. Warrior et al. [40] measured the Mode-I fracture toughness GIc of 0/30/30 braided vinyl ester/carbon composite and reported an average GIc value of 1062 J/m2. The Mode-II experiment was not successful on braided carbon composites [40]. Therefore the Mode-II fracture toughness GIIc for CoFRM, a braided glass/ polyester composite, was used as an estimate. Warrior et al. [39] reported a GIIc value of 2800 J/m2 for CoFRM. DCB and ENF simulations yielded comparable GIc and GIIc values with a NFLS of 12 MPa and a SFLS of 36 MPa. 6.3. Simulation results
Fig. 7. (a) FE model for composite tube crush with a plug initiator. The drop weight and the plug were modeled as rigid and (b) details of the plug initiator.
Initial crack
The tube crash simulations were conducted on vinyl ester/ carbon Fortafil tri-axial 0/+30/30 braid 50 50 mm2 square tubes of 1-ply, 2-ply and 4-ply. For tubes with a plug initiator, simulations were also conducted for tubes made of Fortafil/Grafil tri-axial 0/+45/45 and 0/+60/60 braids assuming the same damage parameters. Table 3 gives a summary of the notation for the tube configurations used in this study. For tubes with a plug initiator, steady axial crash simulations were achieved and the predicted axial crash responses of composite tubes of different materials and configurations were compared well with the test results. Fig. 9 presents the force–displacement trace and the tube deformation modes obtained by simulation at crash distances of 50 and 175 mm for F/G[0/760]2. The delamination between the two plies and the steady crash behavior were in good agreement with the test. Modeling delamination using multi-shell layers with tiebreak contact interface appeared to be a valid approach. In simulations of tube crash without a plug initiator, the composite tube tended to buckle shortly after the crash zone
Crack propagates through delamination.
Fig. 8. FE model for the double cantilever beam (DCB) experiment: (a) initial condition and (b) during simulation.
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with plug
25
Force (kN)
20 15 10 test D0003 test D0031 Simulation
5 0 0
50
100
150 200 Displacement (mm)
250
300
Fig. 9. Comparison of the force–displacement curves of a simulation with two tests for a F/G[0/760]2 tube with a plug initiator.
Force (kN)
Fig. 10. Simulation at a crash distance of (a) 30 mm and (b) 60 mm for a F[0/730]2 tube without a plug initiator.
90 80 70 60 50 40 30 20 10 0 -10
2-ply
0
10
20
30
40 50 Displacement (mm)
60
70
80
Fig. 11. Comparison of the force–displacement curves of a simulation using LLNL data with two tests for a F[0/730]2 tube without a plug initiator.
moved beyond the bevel. This deformation mode was not observed in the tests. Fig. 10 shows the deformation of a F[0/ 730]2 tube at a crash distance of 30 and 60 mm. Global buckling is clearly visible at a crash distance of 60 mm. The force–displacement trace for F[0/730]2 is presented in Fig. 11. Simulations predicted an initial peak force value within 20% for F[0/730]2. However, the simulations predicted a much lower crash force at larger displacement. The overall comparisons between the simulations and test results are presented in Figs. 12 and 13 for the peak crash force and the specific energy absorption, respectively. The predicted peak forces are within 20% of test value. The predictions for the SEA values are generally lower, about 10–20% lower for tubes with
a plug initiator and 30–40% lower for tubes without a plug initiator. The lower prediction for the SEA might be attributed to the deficiency in the modulus reduction law used in MAT58, as discussed earlier. The lower modulus upon unloading results in an over-estimated elastic energy release and hence an under predicted energy dissipation capacity of the material. The lower modulus upon reloading may change the load distribution and affect the deformation mode of the structure. In tube crash, the axial stress at crash front can reach the maximum strength and then reverse sign. Fig. 14 shows a typical axial stress history trace and internal energy variation for an element at the crash front. The element first experienced compressive stress. As the crash
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120
Peak Force (kN)
100
747
test MAT58
80 60
With plug initiator
40
Without plug initiator
20 0 F [0/+-30]
F/G [0/+-45]2 F/G [0/+-60]2
F [0+-30]
F [0+-30]2
F [0+-30]4
Fig. 12. Comparison of simulation and test results for the initial peak force. Predictions are within 20%.
35 30
test MAT58
SEA (J/g)
25 20 15 10 With plug initiator
5 0
F [0/+-30]
Without plug initiator
F/G [0/+-45]2 F/G [0/+-60]2
F [0+-30]
F [0+-30]2
F [0+-30]4
Fig. 13. Comparison of simulation and test results for the specific energy absorption (SEA). Predictions are 10–20% lower for tubes with a plug initiator and 30–40% lower for tubes without a plug initiator.
0.15
axial stress internal energy
Stress (GPa)
0.1 0.05 0 -0.05 -0.1 -0.15 0
0.2
0.4 0.6 Time (ms)
0.8
1
Fig. 14. Element stress and internal energy history plots during an axial crush simulation of braided composite tube.
frond formed and rolled up the axial stress in the element reversed sign from compression to tension, similar to the LLNL compression–tensile test in Fig. 4. This effect is more significant for tubes without a plug initiator where the axial stress is higher and the material is subjected to more severe damage and
consequently suffers a greater modulus reduction before load reversal. Besides the material model, the approximations used in the current work may also impede the accuracy of the simulations. Firstly, the Mode-II interlaminar fracture toughness used in
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Fig. 15. Cross section of Fortafil square tubes: (a) F[0/730]2 and (b) F[0/730]4.
estimating the shear failure strength of tiebreak contact in this work was a value borrowed from a glass/polyester composite. Secondly, the friction coefficient between the delaminated plies was unknown and assumed to be 0.5 for static and 0.35 for dynamic. The friction coefficient between the composites and steel was assumed to be 0.22, though this value is very close to 0.24, the measured dynamic coefficient of friction between braided carbon composite tube and steel trigger recently reported by Brimhall [41]. Lastly, composites are known for their superior damping capacity. So far, the damping factor for braided composites and its dependence on frequency are unknown and the material damping was not considered in these simulations. Another source of uncertainty in crashworthiness prediction of composite tubes is the much greater variations in material properties as well as in tube geometries as a result of low cost and fast processing requirements for automotive composites. The LLNL report [32] revealed that the compressive strength of the braided composite varied as much as 100% depending on the quality of the plaques. As mentioned earlier, Fortafil tubes had a nominal ply thickness of 2.0–2.4 mm, a total variation of 20%. For the same tube, the thickness varies across the tube wall. Fig. 15 shows the cross section of F[0/730]2 and F[0/730]4 tubes. It is visible that the lower left and upper right corners of the F[0/730]4 tube are thinner than their opposite corners. Assuming the fiber content remains the same, the fiber volume fraction will vary at these areas and so do the mechanical properties.
7. Conclusions The constitutive responses of a composite damage material model, MAT58, in LS-DYNA were investigated using a single element test. The results showed that MAT58b, a modified version of Matzenmiller’s damage mechanics model, provides a better representation of the pre- and post-peak responses of composites. Axial tube crush simulations were carried out for Fortafil 0/30/ 30 tri-axial braided tubes. The inter-ply delamination of multilayer tubes was modeled using FE models of multi-shell layers with tiebreak contact interface. The delamination modeling was successful. The predicted peak forces were within 20% of the experimental values but the predicted average crush forces and SEA were generally lower than experimental results. This is likely due to a deficiency in MAT58 in modeling the subsequent unloading response of partially damaged material.
Acknowledgements This work is performed for the Energy Management Working Group (EMWG) of the Automotive Composite Consortium (ACC) under the project of Analytical Tools Development, sponsored by the US Department of Energy, Office of Transportation Technologies, Office of Advanced Automotive Technologies, Lightweight Materials Program under Cooperative Agreement No. DE-FC0595OR22363. The authors would like to thank R. Jeryan of Ford Motors Company, V. Agaram of Daimler Chrysler for their support to this work, Ruth Gusko for the tube crash test, Dilip Bhalsod of LSTC for technical support regarding the use of LS-DYNA. Special thanks to S. DeTeresa of Lawrence Livermore National Laboratory for the permission to use their experimental results on braided composites. References [1] Hull D. Energy absorbing composite structures. Science and Technology Review, University of Wales 1988;3:23–30. [2] Thornton PH, Jeryan R. Crash energy management in composite automotive structures. International Journal of Impact Engineering 1988;7: 167–80. [3] Schmeuser DW, Wickliffe LE, Mase GT. Front impact evaluation of primary structural components of a composite space frame. In: The seventh international conference on vehicle structural mechanics. SAE, Detroit, 1988. p. 67–75. [4] Frutiger RL, Baskar S, Lo KH, Farris R. Design synthesis and assessment of energy management in a composite front end vehicle structure, In: Proceedings of the fifth annual ASM/ESD ACCE conference, Detroit, 1989. p. 33–43. [5] Botkin M, Fidan S, Jeryan R. Crashworthiness of a production vehicle incorporating a fiberglass reinforced composite front structure. SAE paper No. 971522, 1997. [6] Botkin M, Browne AL, Johnson N, Fidan S, Jeryan R, Mahmood H, Wang HR, Lalik L, Peterson LD. Development of a Composite Front Structure for Automotive Crashworthiness, ASME—Crashworthiness, Occupant Protection and Biomechanics in Transportation Systems. ASME Publication AMD-vol. 230/BED-vol. 41, 1998. [7] Hallquist J, Lum LCK, Matzenmiller A. Numerical simulation of post-failure crash of composite tubular beams, ACC technical report, RE EM91-02. [8] Botkin M, Johnson N, Hallquist J, Lum LCK, Matzenmiller A. Numerical simulation of post-failure crashing of composite tubes, In: The second international LS-DYNA3D Conference, September, 1994. [9] Johnson N. ACC generic tube crush program: dynamic crush test of SRIM tubes, In: Proceedings of the American Society for composites 13th technical conference, September, 1998. [10] Johnson N, Browne AL, Houston D, Lalik L. ACC generic tube crush program: resin transfer molded tubes, ASME—Crashworthiness, occupant protection and biomechanics in transportation systems. ASME Publication AMD-vol. 230/BED-vol. 41, 1998. [11] Johnson N, Browne AL, Houston D, Lalik L. ACC generic tube crush program: comparison of axial crush of RTM and SRIM tubes. In: Proceedings of the American society for composites 14th technical conference, 1999. p. 641–50.
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[12] Browne AL, Johnson N. Short apron drop tests in support of automotive composites consortium focal project I, ASME—Crashworthiness, Occupant Protection and Biomechanics in Transportation Systems, ASME Publication AMD-vol. 237/BED-vol. 45, 1999. [13] Browne, AL, Botkin M. Dynamic axial crush of automotive rail-sized composite tubes part 4: plug vs. non-plug crush initiators. In: Proceedings of ASME 2003 international mechanical engineering congress and exposition. [14] Browne AL, Zimmerman KB. Dynamic Axial Crush of Automotive Rail-Sized Composite Tubes Part 2: Tubes with Braided Reinforcements (Glass, Carbon, and Kevlars) and Non-Plug Crush Initiators. In: Proceedings of ASME 2002 international mechanical engineering congress and exposition. [15] Browne AL, Johnson N, Houston D, Lalik L. Automotive composites consortium generic tube crush program: processing and molding effects on the axial crush of RTM and SRIM tubes. In: Proceedings of SAMPE–ACE–DOE advanced composites conference. Detroit, MI, 1999. [16] Botkin M, Johnson N, Zywicz E, Simunovic S. Crashworthiness simulation of composite automotive structures. In: 13th Annual Engineering Society of detroit advanced composite technology conference and exposition. Detroit, 1998. [17] Agaram V, Bilkhu S, Fidan S, Botkin M, Johnson N. Simulation of controlled failure of automotive composite structure with LS-DYNA3D. Advanced composite conference and exhibition. Detroit, April 1997. [18] Xiao X, Johnson N, Botkin M. Challenges in composite tube crash simulation. In: Proceedings of the American Society for composites. 18th Technical conference. October 2003. [19] World wide failure exercise on failure prediction in composites, a special edition. Composite Science and Technology 2002;62. [20] Hinton MJ, Kaddour AS, Soden PD. A comparison of the predictive capabilities of current failure theories for composite laminates, judged against experimental evidence. Composite Science and Technology 2002;62:1725–97. [21] Davies GAO. Structural impact and crashworthiness, vol. 1. London: Elsevier Applied Science Publisher; 1984. [22] Mamalis AG, Manolakoos DE, Demosthenous GA, Ioannidis MB. Crashworthiness of composite thin-walled structural components. Lancaster, PA: Technomic Publishing Co. Inc.; 1998. [23] Beard S, Chang F. Design of braided composites for energy absorption. J Thermoplastic Composite Materials 2002;15(1):3–12. [24] Beard S, Chang F. Energy absorption of braided composite tubes. Int J Crashworthiness 2002;7(2). [25] Flesher ND. Crush Energy Absorption Of Braided Composite Tubes. Ph.D. Dissertation, Stanford University, 2006.
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[26] Quek SC, Waas AM, Shahwan KW, Agaram V. Compressive response and failure of braided textile composites: Part 1—experiments. Int J Non-Linear Mechanics 2004;39:635–48. [27] Quek SC, Waas AM, Shahwan KW, Agaram V. Compressive response and failure of braided textile composites: Part 2—computations. Int J Non-Linear Mechanics 2004;39:649–63. [28] Song S, Waas AM, Shahwan KW, Xiao X, Faruque O. Briaded textile composites under compressive loads: modeling the response, strength and degradation. Composites Science and Technology 2007;67:3059–70. [29] Williams K, Vaziri R, Floyd A, Poursartip A. Simulation of damage progression in laminated composite plates using LS-DYNA. In: The fifth international LSDYNA users conference. Southfield, Michigan, September 21–22, 1998. [30] Williams K, Vaziri R. Application of a damage mechanics model for predicting the impact response of composite materials. Computers and Structures 2001;79:997–1011. [31] Kongshavn I, Poursartip A. Experimental investigation of a strain-softening approach to predicting failure in notched fibre-reinforced composite laminates. Composites Science and Technology 1999;59:29–40. [32] DeTeresa SJ, Allison LM, Cunningham BJ, Freeman DC, Saculla MD, Sanchez RJ, Winchester SW. Experimental results in support of simulating progressive crush in carbon-fiber textile composites. Lawrence Livermore National Laboratory, UCRL-ID-143287, March 12, 2001. [33] LS-DYNA Keyword User’s Manual, Version 950. [34] Matzenmiller A, Lubliner J, Taylor RL. A constitutive model for anisotropic damage in fiber-composites. Mechanics of Materials 1995;20:125–52. [35] Schweizerhof K, Weimar K, Munz Th, Rottner Th. Crashworthiness analysis with enhanced composite material models in LS-DYNA—merit and limits. In: Proceedings of the fifth international LS-DYNA conference. Southfield, Michigan, 1998. [36] Browne Alan L, Johnson Nancy L. Dynamic crush tests using a ‘‘free-flight’’ drop tower: theory. Experimental Techniques 2002;26(5):43–6. [37] Hallquist JO. LS-DYNA Theoretical Manual, 1998. [38] Daniel IM, Ishai O. Engineering mechanics of composite materials. New York: Oxford University Press; 1994. [39] Warrior NA, Bailey DA, Ducket MJ, Fernie R, Lourenc- o N. Rate dependent effects on crash zone morphology of polymer composites. Technical Progress Report 4, October 1999. [40] Warrior NA, Ducket MJ, Fernie R, Bailey DA, Lourenc- o N. Rate dependent effects on crash zone morphology of polymer composites. Technical Progress Report 7, July 2000. [41] Brimhall TJ. Friction energy absorption in fiber reinforced composites. PhD dissertation, Michigan State University, May 2005.
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Axial crush simulation of braided carbon tubes using MAT58 in LS-DYNA Xinran Xiao , Mark E. Botkin, Nancy L. Johnson General Motors Corporation, R&D Center, MC 480-106-224, 30500 Mound Road, Warren, MI 48090-9055, USA
a r t i c l e in f o
a b s t r a c t
Article history: Received 17 October 2007 Accepted 14 December 2008 Available online 4 February 2009
This paper examines a composite damage constitutive model, MAT58, in LS-DYNA and its application for use in braided composite tube axial crush simulations. The constitutive response of MAT58 was investigated using single element simulations. It was found that MAT58 reproduced the softening behavior of the braided composite under monotonic compressive loading, but failed in subsequent unloading and tensile loading cycles. A deficiency in the damage law in MAT58 was identified. Unloading and reloading a volume of material that had suffered some degree of damage was a part of the process with the progressing of crush zone during the axial crush of composite tubes. Consequently, this deficiency hinders the success of MAT58 in such applications. In tri-axial braided composite tube axial crush simulations, although the predicted initial peak forces were within 20% of the experimental values, the predictions for the specific energy absorption (SEA) values were consistently low, particularly for tubes without a plug as crush initiator. These discrepancies are attributable to the deficiency in the damage law in MAT58. & 2008 Published by Elsevier Ltd.
Keyword: Carbon fiber composites Dynamic finite element simulation Damage Crashworthiness Tube crush Energy absorption
1. Introduction The exceptionally high specific energy absorption (SEA) of fiber reinforced polymer composites [1] had long captured the imagination of automotive design engineers at Detroit [2–4]. Even though composites are common in primary structural components in racing cars, the challenges in developing automotive production-feasible composite vehicles are formidable. Towards this common goal, the three Detroit-based automakers, Ford, Chrysler and General Motors, formed the Automotive Composite Consortium (ACC) in 1988. One of the challenges in structural requirements for a passenger vehicle is the crashworthiness requirement. ACC has demonstrated in its first focal project (FP-I) that composite front structures can meet the crash safety requirements [5,6]. In FP-I design, the crash response and crash force of the front rails were determined by using a lumped-mass model with vehicle dynamic analysis [6]. The information was then fed into a finite element collapsible beam code to determine the load and section properties. Static finite element analysis was employed to determine the stress field. Even though crash simulations using dynamic finite element analysis (FEA) had become routine work in the automotive industry during the course of FP-I program, its applications to composite structures were not successful. Since then, the ACC has been working on developing predictive tools for crash simulation of composite structures [7,8]. In crash
Corresponding author. Tel.: +1 517 884 1606; fax: +1 517 884 1601.
E-mail address: [email protected] (X. Xiao). 0263-8231/$ - see front matter & 2008 Published by Elsevier Ltd. doi:10.1016/j.tws.2008.12.004
energy management, one critical component of the front structure is the front rail. Extensive experimental studies have shown that braided composite tubes are one of the most promising geometries for the front rail structures [9–15]. Axial crush simulation of braided composite tubes has become a benchmark for the predictive capability of composite crash simulation [7,8,16–18]. It turns out that the crush simulation of composite tubes is a rather difficult task and the progress in the area has been slow. Actually, even under quasi-static loading, failure prediction in laminated composites remains a challenging task. To gauge the predictive capability of laminate composite failure theories, a ‘‘World Wide Failure Prediction of Composite Laminate Exercise’’ (so-called ‘‘Failure Olympics’’) was organized, which involved more than a dozen experts and lasted over 10 years [19]. The final results showed that no one theory could predict all 14 cases given for the exercise although their success rates varied a great deal [20]. The prediction of the composite tube crush is certainly much more challenging since one is dealing with the initial failure as well as the post-peak behavior under dynamic loading and in more complicated material forms. Unlike design analysis where a maximum load, a limiting deformation value, or a structural change satisfying a set of load conditions is the ultimate goal, the vehicle crashworthiness design is aimed at maximizing the energy absorption of certain components while minimizing deformation in the passenger compartment [21,22]. This is achieved by controlled progressive deformation and fracture, load sharing and load transfer between components during deformation. In other words, crash simulation of composites requires the modeling of post-peak response of the materials.
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Post-peak analysis is the key in composite tube crush simulations. After reaching a peak stress at the maximum strength, fabric composites such as braids can undergo large deformations at lower stresses before completely losing the load carrying capability. Local damages certainly have significant impact on the load carrying capacity along the principal stress direction while the load carrying capacity in other directions may or may not suffer. Tube crush also invokes multi-axis stresses in materials. In addition, the strain rate effect of the material may not be negligible. To simulate composite tube crush, one needs constitutive models that can represent these behaviors reasonably well up to the complete fracture of a material. At the same time, one must be able to handle the numerical problems associated with large deformation and fracture. These include element distortion, contact, hourglassing and instability. It is not uncommon for a composite tube crush analysis to abort before completion due to these numerical difficulties. Over the years the ACC has worked with national labs, university researchers and software companies to develop computational tools for the crashworthiness prediction of composite tube structures. The earliest development traced back to 1990 when the ACC sponsored Livermore Software Technology Corporation (LSTC) to implement a composite damage model, MAT58a, in explicit LS-DYNA [7]. The existing constitutive models and the models under development for composite crash application may be classified into two major categories: micro-mechanics models and phenomenological models. A true micro-mechanics model captures the deformation, damage and fracture of individual fiber yarns, matrix, their interface and interaction in a unit cell based on the constituent properties. The model predicts the overall stress–strain response of the unit cell as damage progresses, which is then used in the global model for the element. In reality it is difficult to model every type of damage and its subsequent softening effects in a complicated 3D microstructure. The problem has been approached by adopting simplifications and assumptions at the unit cell level [23–25] or by detailed large-scale modeling [26–28]. Beard and Chang [23,24] developed a micro-mechanics-based model. It calculates the constitutive behavior of a unit cell through an idealized three-dimensional description of the braid geometry using fiber tow and resin properties. The model includes the effect of tow scissoring in braids and considers the damage and fracture in constituents using the maximum stress criterion. It updates the effective stiffness of the unit cell as damage progresses. The model has been implemented in implicit [23] and explicit [25] ABAQUS and used for analysis of tube crush with a plug initiator. Quek et al. [26,27] and Song et al. [28] developed a detailed finite element (FE) model for a representative unit cell (RUC) in a braided composite. The RUC predicted softening responses for tri-axial braided composites under compression. Introducing imperfection into the RUC further improved correlations between simulations and experimental results. Phenomenological models, on the other hand, describe the global constitutive behavior. In fact, most of the material models in FE codes are phenomenological models. Commonly used piecewise linear plasticity model for metallic materials is a good example. Instead of modeling plastic deformation resulted from a complicated set of deformation mechanisms such as the motion of individual dislocations, their interactions with barriers such as grain boundaries and precipitations, and interaction between dislocations, etc., the plasticity model describes the nonlinear deformation through a stress–strain relation measured at the macroscopic level. The continuum damage mechanics-based phenomenological models are characterized by the use of damage variables and
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damage evolution laws. The damage variables are internal variables that cannot be measured readily by standard material tests. The code developers often provide insufficient information about how to obtain these parameters. Consequently the parameters in damage models in FE codes are determined by curve fitting, so-called calibration. This is further complicated by the fact that the standard composite material tests generally do not provide information beyond the peak load. Even though some composite structures, such as tubes made of fabrics, exhibit significant load carrying capacity after initial peak load, such behaviors were not recorded in material tests. To model the crash responses of these composite structures, the material must have some post-peak load carrying capabilities. Therefore, various forms of post-peak behaviors were assumed. The damage parameters were selected by best fit to a component test curve. The problem is that a set of parameters obtained under one condition often does not work for the next condition. This prohibits the predictive capability of the analysis. Physics-based composite damage models have been developed for laminated composites and implemented in commercial dynamic FE codes. The CODAM model developed by Williams et al. [29,30] is an example. An oversized CT specimen was used to generate specimens with different levels of damage to obtain the parameters in CODAM needed for post-peak response under tensile load [31]. A proper material testing method is the key in developing physics-based phenomenological models. To authors’ best knowledge, the work by DeTeresa et al. [32] at the Lawrence Livermore National Lab (LLNL) was the first to generate damage data using un-notched specimens as in a material test on fabric composites. Using LLNL data in MAT58 of LS-DYNA, we obtained encouraging results in recent attempts in tube crush simulations. The results of this new initiative are summarized in this paper.
2. Composite material model—MAT58 MAT58 in LS-DYNA [33] was based on Matzenmiller’s damage mechanics model [34]. The model has the capability of modeling the damage independently in the principle directions of orthotropic materials. For the plane stress condition, the constitutive equation is given as [7]: 2 3 3 2 3 2 s1 1 ð1 o1 Þð1 o2 Þn21 E2 0 ð1 o1 ÞE1 6 7 16 7 6 7 6 s2 7 ¼ 6 ð1 o1 Þð1 o2 Þn12 E1 7 7 6 ð1 o ÞE 0 2 2 5 4 2 5 4 4 5 c
0
t
0
c ¼ 1 ð1 o1 Þð1 o2 Þn12 n21
cð1 os ÞG
g
(1)
In Eq. (1), s is the normal stress, t is the shear stress, e is the normal strain, g is the shear strain, E is the modulus, G is the shear modulus, n is Poisson’s ratio, and o is the damage function. As a special case, an exponential damage evolution law was provided mi
oi ¼ 1 eði =0i Þ
(2)
with i ¼ 1T, 1C, 2T, 2C, S (T ¼ tensile, C ¼ compressive, S ¼ shear). In Eq. (2), mi and e0i, are sets of parameters. e0i is related to other parameters and properties by
0i ¼
X t;c ðmi eÞ1=mi Ei
(3)
where X is the strength. The parameter mi controls the shape of the stress–strain response. Fig. 1 shows the stress–strain curves for different mi values. Eq. (3) was implemented in MAT58 of LS-DYNA in its earlier version [23], MAT58a.
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1.2
where a is the ratio of the limiting stress to the peak stress, 0oap1.0. a ¼ 1.0 corresponds to no strain softening. Eq. (4) gives the user an option to avoid the localization. Typical stress–strain curves for various values of Ei and SLIMi are shown in Fig. 2.
1 m = 0.25 m = 0.5 m=1
0.6
3. Lawrence Livermore National Lab (LLNL) test data
m=2 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
Strain Fig. 1. Examples of stress–strain responses for various m values, MAT58a.
1.2 1.0
Stress
0.8 0.6 0.4 E11t = 0.01, slimt1 = 0.5 E11t = 0.02, Slimt1 = 0.5
0.2
E11t = 0.03, Slimt1 = 0.7 Elastic
0.0 0
0.02
0.04
0.06
0.08
0.1
Strain Fig. 2. Examples of stress–strain responses for various Ei and SLIMi values, MAT58b.
One of the shortcomings of MAT58a is that not only the postpeak response but also the pre-peak response varies with the mi value. As shown in Fig. 1, to simulate a material with relatively large post-peak deformation, one needs to select a smaller mi value to obtain a slower unloading curve. A smaller mi, however, results in a softer pre-peak response, which deviates significantly from the nearly elastic behavior exhibited by common continuous fiber reinforced composites. The direct coupling between the preand post-peak stress–strain relation compels the analyst to sacrifice one response to satisfy the other. Later, the damage evolution law in MAT58 was modified [35]. The modified version is MAT58b. In MAT58b mi is not employed. Instead, two new sets of parameters Ei (i ¼ 11T, 11C, 22T, 22C, S) and SLIMi (i ¼ T1, C1, T2, C2, S) are introduced. Ei is defined as the strain at the maximum stress response and SLIMi as the minimum stress limit of damaged material. A parameter study indicated that varying Ei changed the slope of the pre- and post-peak response, the same way as mi in MAT58a except that Ei appears to be reciprocal to mi, i.e. a greater Ei value results in a smaller slope of the stress–strain response. SLIMi sets a predefined limiting stress. At the limiting stress, the damage law is described by [35]
o¼1
aX t;c E
(4)
To support the constitutive model development for composite tube crush applications, DeTeresa et al. [32] measured the socalled ‘‘total stress–strain responses’’ of a braided carbon composite under tensile and compressive loading. The flat plaques used in the study were 0/730 tri-axial braid manufactured using resin transfer molding. The braid contains 80 k carbon fiber tows in both axial and bias direction. The matrix is an epoxy vinyl ester resin. Both tensile and compression tests were conducted using straight-sided specimens. The tensile test was similar to a standard test except the specimens were pulled until complete fracture (zero load) was attained. For the compression test, rather than using a standard tabbed test specimen and a compression fixture for fiber composites, untabbed specimens were gripped directly with hydraulic grips in a servo hydraulic test machine. With a short gage length, acceptable compressive failure was achieved. An apparent strain was calculated from the machine displacement and the specimen gage length. The specimens were compressed well beyond the initial peak load until about 15–20% strain. Fig. 3 plots a set of compressive stress–strain curves measured in [32]. These data reveal that the braided composite had retained a significant fraction of its load carrying capacity after having reached the commonly defined failure strength, particularly under compressive loading. The post-peak responses of this composite resemble those depicted by MAT58b in Fig. 2, suggesting that the parameter SLIMi may be estimated from the total stress–strain curves. DeTeresa et al. [32] also measured the total stress–strain curves for off-axis angles and 901 specimens at an intermediate strain rate of 1/s. They reported that the strengths were about 30% higher than those measured in quasi-static tests. The strains at maximum stress were also higher when compared with the quasistatic values. The modulus was not measured in the strain rate test. Table 1 provides the mechanical properties of the tri-axial braided composites. Table 2 lists the subset of data relevant to that needed for MAT58 extracted from the results of [32]. It shows
16 14 Compression Stress (ksi)
Stress
0.8
12 10 8 6 4 2 0
0
0.05 0.1 Apparent Compression Strain (in./in.)
0.15
Fig. 3. Compressive stress–strain response of triaxial braided panel specimens in 01 direction, 4 repetitions [32].
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Table 1 Mechanical properties of triaxial braid 0/+30/30, Fortafil. Mechanical properties
Ultimate strength (MPa)
(Chord) Modulus (GPa)
Strain to failure
Poisson’s ratio
01 tensile 901 tensile 01 compression 901 compression
305.4 32.1 120.7 66.3
42.3 5.7 32.2 8.6
0.0104 0.0097 0.0039 0.0085
1.169 0.207
Table 2 Data extracted from DeTeresa et al. [32] for MAT58 application. Type of test
Strain rate
Maximum stress (MPa)
Ei, strain at maximum stress
SLIMi, ratio of min. stress/max. stress
01 tensile 901 tensile 901 tensile 01 compression 901 compression 901 compression
Quasi-static Quasi-static 1/s Quasi-static Quasi-static 1/s
299.2 24.1 38.0 93.7 56.9 71.7
0.01 0.009 0.02 0.01 0.014 0.025
0.05 0.57 0.43 0.44 0.42 0.34
60 40 20
Stress (MPa)
0 -20 -40 -60 -80 -100 LLNL test MAT58
-120 -140 -0.06
-0.04
-0.02
0
0.02 Strain
0.04
0.06
0.08
Fig. 4. Comparison of the MAT58 simulation to LLNL compression–tension test data.
that for the braided composite tested at LLNL, the SLIMi values are between 0.3 and 0.5. The 01 tensile test is an exception, which yielded a low SLIMi value about 0.05.
4. Single element test In the LLNL report, DeTeresa et al. [32] presented the results of a compression–tension test. The specimen was loaded first in compression beyond the onset of failure and then reversed to tension. The recorded stress–strain lotus is plotted in Fig. 4. DeTeresa et al. noted that the compressively damaged material did not sustain a tensile load until most of the compressive displacement was recovered. A single element model was used to examine whether MAT58 could reproduce the stress–strain lotus of the LLNL test. In the simulation, the element was compressively loaded through prescribed nodal displacements to the same maximum strain level as in the test and then the direction of the displacement was
Fig. 5. Tube crush experiment setup.
reversed. The element stress–strain lotus is compared with the test data in Fig. 4. As seen, the stress–strain response obtained by simulation matched the test curve reasonably well over the compressive loading segment but poorly over the unloading and tensile loading segments of the curve. Matzenmiller’s damage mechanics model [34] treats the softening behavior through modulus reduction. This treatment is computationally efficient, easy for implementation in FE codes and is also sufficient to represent softening behaviors of damaged materials under monotonic loading. To predict the energy absorption, however, one needs to model not only the loading but also unloading response of damaged materials. The elastic modulus in unloading segment affects the elastic energy released by the material. Because the energy conservation is sustained in dynamic analysis, an over-predicted elastic energy release changes the energy
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Table 3 Notation for tube configurations. Notation
Tube configuration
F[0/730] F[0/730]2 F[0/730]4 F/G[0/730]2 F/G[0/745]2 F/G[0/760]2
1-ply 2-ply 4-ply 2-ply 2-ply 2-ply
triaxial triaxial triaxial triaxial triaxial triaxial
braid, braid, braid, braid, braid, braid,
0/+30/30, 0/+30/30, 0/+30/30, 0/+30/30, 0/+45/45, 0/+60/60,
Fortafil Fortafil Fortafil Fortafil Fortafil Fortafil
80 k 80 k 80 k 80 k 80 k 80 k
tows tows tows axial tow, Grafil 12 k bias tows axial tow, Grafil 12 k bias tows axial tow, Grafil 12 k bias tows
distribution of the system. The released elastic energy will be fed back into the system causing a sudden increase in kinetic energy and hence a greater tendency of instability of the system.
5. Tube crush test The axial crush experiments of the composite tubes were conducted using a drop tower [36]. Fig. 5 shows a tube crush experiment setup. The tube was mounted to the drop head and raised to the predetermined height and then the drop head was released to fall under gravity. The tube was crushed against a steel plate. The force was calculated from the recording of four load cells positioned under the steel plate. The velocity and displacement were calculated from the signal of an accelerometer attached to the drop head. The data acquisition system had a sampling rate of 20 kHz, i.e. an interval of 50 ms. For the crush experiments reported in this paper, the total drop mass was 140 kg and the drop height was 2.54 m. The recorded initial velocity was 7.1 m/s. The composite tubes were 360 mm in length. To ensure progressive failure, the crush end of the tube was chamfered with a 451 bevel. The tubes considered in this study had a square cross section of 55 55 mm2. Two sets of composite tubes were considered in this work. The first set was 0/+30/30 tri-axial braided 50 50 mm2 square tubes. Fortafil 80 k carbon fiber tows1 were used as axial and bias tows. The resin was epoxy vinyl ester resin. The tubes were made of 1-ply, 2-ply and 4-ply of braids. These tubes will be identified as F[0/730], F[0/730]2 and F[0/730]4 hereafter in the text. Table 3 gives a summary of the notation for the tube configurations used in this study. Composite tubes were manufactured using the resin transfer molding (RTM) process [10]. The average thickness of the molded composites varied between 2.0 and 2.4 mm per ply. Flat panels of the tri-axial braided composite were also manufactured. The mechanical properties of the composite were tested using specimens cut from flat panels. The second set of tubes included 2-ply 50 50 mm2 tubes of Fortafil/Grafil tri-axial braid of 0/+30/30, 0/+45/45 and 0/+60/ 60 configurations, i.e. F/G[0/730]2, F/G[0/745]2 and F/G[0/ 760]2. In these tubes Fortafil 80 k fiber tows were used as the axial tows and Grafil 12 k fiber tows were used as bias tows. The tubes were manufactured using RTM process. The smaller bias tows resulted in a thinner tube wall. Fortafil tubes yielded an average wall thickness of 2–2.4 mm per ply. Fortafil/Grafil tubes had about 1.1–1.2 mm per ply. Some tubes were tested with a metal plug-type initiator inserted at the beveled end of the tube. Fig. 6 shows a tri-axial braided carbon tube after crush testing and the initiator used for 55 55 mm2 square tubes. The initiator had a 6 mm active radius. 1
80 k tow ¼ 80,000 fiber filaments in a tow.
Fig. 6. Tri-axial carbon tube after crush test.
The plug was attached to the tube using hot glue. In addition, a rubber pad was attached to the bottom of the plug.
6. Tube crush simulation 6.1. Finite element models The FE model consists of three portions: the drop weight, the composite tube and a plug-type initiator for tests with a plug initiator or a rigid wall for tests without a plug initiator. Fig. 7 shows the FE model for a tube with a plug initiator. The drop weight was modeled as rigid with an equivalent weight. The composite tube was modeled using shell elements. The tube was joined to the drop weight through constrained nodes. An initial velocity of 7.1 m/s was assigned to the drop weight and the tube. For tube crush simulations without a plug initiator, the rigid wall force was monitored to compare with the experimental result. The displacement in simulation was taken from the nodal displacement at the center of the upper surface of the drop weight. For tube crush simulations with a plug initiator, the Z-component of the interface force between the tube and the plug was monitored to compare with the force measured in the experiment. The plug-type initiator was modeled as rigid and its lower surface was fully constrained. The multi-layered composite tubes were modeled with multi layers of shell elements and the shell layers were connected through tiebreak contact definition. Details about the tiebreak definition will be discussed later.
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The bevel was modeled with a single row of shell elements per layer. The elements were assigned a tapered thickness such that the two nodes at the leading edge had a smaller shell thickness of 0.1 mm. In the later simulations the bevel was modeled with two rows of shell elements. This change led to a more gradual rise of the initial force response. Three element formulations, Belytschko–Tsay, Belytschko– Wong–Chiang and fully integrated shell [37], were investigated. The more computationally expensive elements did not produce notable improvement to the results and hence, the default element formulation, Belytschko–Tsay, was employed for the analysis. Tube crush simulations were tested with different numbers of through thickness integration points. It was observed that increasing the number of integration points from two to three slightly improved the smoothness of the simulation but added to the computation time. Increasing the number of integration points beyond three did not lead to further improvement. In the end, two integration points were employed in simulations.
6.2. Delamination modeling Delamination was one of the major damage mechanisms in axial crush tests of multi-layered composite tubes. To allow delamination failure, the tube was modeled with multi-layer shell elements connected through the tiebreak contact definition of LSDYNA [33]. Tiebreak contact is a special type of contact. It works the same as common contact types under compressive load. Under tensile and shear loads, tiebreak allows the separation of the tied surfaces following an interface strength-based failure criterion. The
745
following criterion was used in this work:
jsn j NFLS
2
þ
jss j SFLS
2 X1
(5)
where sn is the normal stress, ss is the shear stress, NFLS is the normal failure stress and SFLS is the shear failure stress. The user can choose a failure criterion in other forms including defining a damage curve that allows the bondline deformation before complete separation of the two surfaces. The NFLS and SFLS values cannot be measured directly. In this work these values were estimated through correlating simulations with interlaminar fracture toughness experiments. The NFLS value was obtained by correlating simulations with the load–displacement traces measured in Mode-I fracture toughness experiment using the double cantilever beam (DCB) [38], similar to the method used by Warrior et al. [39]. Fig. 8 shows the FE model for the DCB specimen. It consists of two layers of shell elements connected through tiebreak contact at the base section. Adjusting the NFLS value, we can match the peak values of the load–displacement curves obtained by simulations with the experimental data. Similarly, the SFLS value was obtained through correlating simulations with the end notch flexural (ENF) experimental results that measures the interlaminar Mode-II fracture toughness. Warrior et al. [40] measured the Mode-I fracture toughness GIc of 0/30/30 braided vinyl ester/carbon composite and reported an average GIc value of 1062 J/m2. The Mode-II experiment was not successful on braided carbon composites [40]. Therefore the Mode-II fracture toughness GIIc for CoFRM, a braided glass/ polyester composite, was used as an estimate. Warrior et al. [39] reported a GIIc value of 2800 J/m2 for CoFRM. DCB and ENF simulations yielded comparable GIc and GIIc values with a NFLS of 12 MPa and a SFLS of 36 MPa. 6.3. Simulation results
Fig. 7. (a) FE model for composite tube crush with a plug initiator. The drop weight and the plug were modeled as rigid and (b) details of the plug initiator.
Initial crack
The tube crash simulations were conducted on vinyl ester/ carbon Fortafil tri-axial 0/+30/30 braid 50 50 mm2 square tubes of 1-ply, 2-ply and 4-ply. For tubes with a plug initiator, simulations were also conducted for tubes made of Fortafil/Grafil tri-axial 0/+45/45 and 0/+60/60 braids assuming the same damage parameters. Table 3 gives a summary of the notation for the tube configurations used in this study. For tubes with a plug initiator, steady axial crash simulations were achieved and the predicted axial crash responses of composite tubes of different materials and configurations were compared well with the test results. Fig. 9 presents the force–displacement trace and the tube deformation modes obtained by simulation at crash distances of 50 and 175 mm for F/G[0/760]2. The delamination between the two plies and the steady crash behavior were in good agreement with the test. Modeling delamination using multi-shell layers with tiebreak contact interface appeared to be a valid approach. In simulations of tube crash without a plug initiator, the composite tube tended to buckle shortly after the crash zone
Crack propagates through delamination.
Fig. 8. FE model for the double cantilever beam (DCB) experiment: (a) initial condition and (b) during simulation.
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with plug
25
Force (kN)
20 15 10 test D0003 test D0031 Simulation
5 0 0
50
100
150 200 Displacement (mm)
250
300
Fig. 9. Comparison of the force–displacement curves of a simulation with two tests for a F/G[0/760]2 tube with a plug initiator.
Force (kN)
Fig. 10. Simulation at a crash distance of (a) 30 mm and (b) 60 mm for a F[0/730]2 tube without a plug initiator.
90 80 70 60 50 40 30 20 10 0 -10
2-ply
0
10
20
30
40 50 Displacement (mm)
60
70
80
Fig. 11. Comparison of the force–displacement curves of a simulation using LLNL data with two tests for a F[0/730]2 tube without a plug initiator.
moved beyond the bevel. This deformation mode was not observed in the tests. Fig. 10 shows the deformation of a F[0/ 730]2 tube at a crash distance of 30 and 60 mm. Global buckling is clearly visible at a crash distance of 60 mm. The force–displacement trace for F[0/730]2 is presented in Fig. 11. Simulations predicted an initial peak force value within 20% for F[0/730]2. However, the simulations predicted a much lower crash force at larger displacement. The overall comparisons between the simulations and test results are presented in Figs. 12 and 13 for the peak crash force and the specific energy absorption, respectively. The predicted peak forces are within 20% of test value. The predictions for the SEA values are generally lower, about 10–20% lower for tubes with
a plug initiator and 30–40% lower for tubes without a plug initiator. The lower prediction for the SEA might be attributed to the deficiency in the modulus reduction law used in MAT58, as discussed earlier. The lower modulus upon unloading results in an over-estimated elastic energy release and hence an under predicted energy dissipation capacity of the material. The lower modulus upon reloading may change the load distribution and affect the deformation mode of the structure. In tube crash, the axial stress at crash front can reach the maximum strength and then reverse sign. Fig. 14 shows a typical axial stress history trace and internal energy variation for an element at the crash front. The element first experienced compressive stress. As the crash
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120
Peak Force (kN)
100
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test MAT58
80 60
With plug initiator
40
Without plug initiator
20 0 F [0/+-30]
F/G [0/+-45]2 F/G [0/+-60]2
F [0+-30]
F [0+-30]2
F [0+-30]4
Fig. 12. Comparison of simulation and test results for the initial peak force. Predictions are within 20%.
35 30
test MAT58
SEA (J/g)
25 20 15 10 With plug initiator
5 0
F [0/+-30]
Without plug initiator
F/G [0/+-45]2 F/G [0/+-60]2
F [0+-30]
F [0+-30]2
F [0+-30]4
Fig. 13. Comparison of simulation and test results for the specific energy absorption (SEA). Predictions are 10–20% lower for tubes with a plug initiator and 30–40% lower for tubes without a plug initiator.
0.15
axial stress internal energy
Stress (GPa)
0.1 0.05 0 -0.05 -0.1 -0.15 0
0.2
0.4 0.6 Time (ms)
0.8
1
Fig. 14. Element stress and internal energy history plots during an axial crush simulation of braided composite tube.
frond formed and rolled up the axial stress in the element reversed sign from compression to tension, similar to the LLNL compression–tensile test in Fig. 4. This effect is more significant for tubes without a plug initiator where the axial stress is higher and the material is subjected to more severe damage and
consequently suffers a greater modulus reduction before load reversal. Besides the material model, the approximations used in the current work may also impede the accuracy of the simulations. Firstly, the Mode-II interlaminar fracture toughness used in
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Fig. 15. Cross section of Fortafil square tubes: (a) F[0/730]2 and (b) F[0/730]4.
estimating the shear failure strength of tiebreak contact in this work was a value borrowed from a glass/polyester composite. Secondly, the friction coefficient between the delaminated plies was unknown and assumed to be 0.5 for static and 0.35 for dynamic. The friction coefficient between the composites and steel was assumed to be 0.22, though this value is very close to 0.24, the measured dynamic coefficient of friction between braided carbon composite tube and steel trigger recently reported by Brimhall [41]. Lastly, composites are known for their superior damping capacity. So far, the damping factor for braided composites and its dependence on frequency are unknown and the material damping was not considered in these simulations. Another source of uncertainty in crashworthiness prediction of composite tubes is the much greater variations in material properties as well as in tube geometries as a result of low cost and fast processing requirements for automotive composites. The LLNL report [32] revealed that the compressive strength of the braided composite varied as much as 100% depending on the quality of the plaques. As mentioned earlier, Fortafil tubes had a nominal ply thickness of 2.0–2.4 mm, a total variation of 20%. For the same tube, the thickness varies across the tube wall. Fig. 15 shows the cross section of F[0/730]2 and F[0/730]4 tubes. It is visible that the lower left and upper right corners of the F[0/730]4 tube are thinner than their opposite corners. Assuming the fiber content remains the same, the fiber volume fraction will vary at these areas and so do the mechanical properties.
7. Conclusions The constitutive responses of a composite damage material model, MAT58, in LS-DYNA were investigated using a single element test. The results showed that MAT58b, a modified version of Matzenmiller’s damage mechanics model, provides a better representation of the pre- and post-peak responses of composites. Axial tube crush simulations were carried out for Fortafil 0/30/ 30 tri-axial braided tubes. The inter-ply delamination of multilayer tubes was modeled using FE models of multi-shell layers with tiebreak contact interface. The delamination modeling was successful. The predicted peak forces were within 20% of the experimental values but the predicted average crush forces and SEA were generally lower than experimental results. This is likely due to a deficiency in MAT58 in modeling the subsequent unloading response of partially damaged material.
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