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Modelling fabric reinforced composites under impact loads
Composites: Part A 32 (2001) 1197±1206
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Modelling fabric reinforced composites under impact loads A.F. Johnson* German Aerospace Center (DLR), Institute of Structures and Design, Pfaffenwaldring 38-40, D-70569 Stuttgart, Germany Received 21 March 2000; revised 6 November 2000; accepted 4 December 2000
Abstract The paper describes recent progress on the materials modelling and numerical simulation of the in-plane response of ®bre reinforced composite structures. A continuum damage mechanics model for fabric reinforced composites under in-plane loads is presented. It is based on methods developed for UD ply materials (Compos. Sci. Technol., 43 (1992) 257), which are generalised here to fabric reinforcements. The model contains elastic damage in the ®bre directions, with an elastic±plastic model for inelastic shear effects. Test data on a glass fabric/ epoxy laminate show the importance of inelastic effects in shear. A strategy is described for determining model parameters from the test data. The fabric model is being implemented in an explicit FE code for use in crash and impact studies and preliminary results are presented on a plate impact simulation. q 2001 Published by Elsevier Science Ltd. Keywords: Composite materials; C. Damage mechanics
1. Introduction Composite materials are now being used in primary aircraft structures, particularly in helicopters, light aircraft, commuter planes and sailplanes, because of numerous advantages including low weight, high static and fatigue strength and the possibility to manufacture large integral shell structures. Materials such as carbon ®bre/epoxy are inherently brittle and usually exhibit a linear elastic response up to failure with little or no plasticity. Thus composite structures are vulnerable to impact damage and have to satisfy certi®cation procedures for high velocity impact from runway debris or bird strike. Conventional metallic structures absorb impact and crash energy through plastic deformation and folding. Modern explicit FE codes are able to model these effects and are being successfully applied to simulate the collapse of metallic aircraft and automotive structures. This paper is concerned with the development and validity of such codes for modelling the response of composite structures under impact loads. This topic is being studied in some detail within a CEC funded research project on `High velocity impact of composite aircraft structures' HICAS [2]. This project includes an extensive composites materials and
* Tel.: 149-711-686-2297; fax: 149-711-686-2227. E-mail address: [email protected] (A.F. Johnson). 1359-835X/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII: S 1359-835 X(00)00 186-X
structures test programme, composites modelling developments, FE code implementation and impact simulations. Two important aspects of impact modelling are delamination, which is important in lower energy impacts and in failure initiation, and in-plane ply failure, which controls ultimate failure and penetration in the structure. This paper summarises some of the modelling developments being carried out at the DLR within the project on in-plane damage models for fabric reinforced composite plies, since fabric reinforcements are of considerable practical interest for aircraft structures. Related work is considering delamination failure models and rate dependent behaviour. Emphasis is given to composite materials models suitable for implementation into FE codes, which can adequately characterise the nonlinear damage progression and different failure modes that occur in fabric plies. A continuum damage mechanics model for fabric reinforced composites under in-plane loads is presented. It is based on methods developed for UD ply materials by LadeveÁze and co-workers in Refs. [1,4], which are generalised here to fabric reinforcements. The model contains elastic damage in the ®bre directions, with an elastic±plastic model for inelastic shear effects. A strategy is then described for determining model parameters from the test data. The fabric model is being implemented into PAMCRASH [5], a commercial explicit FE code, and preliminary results are presented on the simulation of ply damage and failure in plate impact.
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development. It can then be shown that the strains e and the damage forces Y are de®ned from the strain energy function by:
2. Composites fabric ply damage model 2.1. Elastic damage mechanics model The fabric reinforced composite ply is modelled as a homogeneous orthotropic elastic or elastic±plastic damaging material whose properties are degraded on loading by microcracking prior to ultimate failure. A continuum damage mechanics formulation is used [3] in which ply degradation parameters are internal state variables, which are governed by damage evolution equations. General constitutive laws for orthotropic elastic materials with internal damage parameters are described by LadeveÁze [4]. In the plane stress case required here to characterise the properties of composite plies or shell elements with orthotropic symmetry axes (x1, x2), the in-plane stress and strain components are
s s 11 ; s 22 ; s 12 T
e e11 ; e22 ; 2e12 T :
1
Using a strain equivalent damage mechanics formulation, an elastic compliance matrix S is assumed with the general form: 0 1 1=E1 1 2 d1 2n12 =E1 0 B C C 2 SB 1=E2 1 2 d2 0 @ 2n12 =E1 A 0
0
1=G12 1 2 d12
The compliance matrix has three scalar damage parameters d1, d2, d12 and four initial or `undamaged' elastic constants: Young's moduli in the principal orthotropy directions E1, E2, the in-plane shear modulus G12, and the principal Poisson's ratio n 12. The damage parameters have values 0 # di # 1 and represent modulus reductions under different loading conditions due to microdamage in the material. Thus for fabric plies d1 and d2 are associated with damage or failure in the principal ®bre directions, and d12 with in-plane shear failure. In general, they will be invariant functions of suitable physical parameters, which characterise the damage state. In the present model, for simplicity, there is no additional damage parameter to model independent degradation in Poisson's ratios. The effect of this assumption is that as damage progresses n 12 is reduced by the factor 1 2 d1 ; since for a uniaxial stress ®eld s 11 it can be shown from Eqs. (1) and (2) that 2e 22 =e11 n12 1 2 d1 ; Similarly it is found under pure transverse stress s 22 that n 21 is reduced by the factor 1 2 d 2 : These assumptions are considered reasonable for the ®rst order damage evolution model described in this paper. The general damage mechanics formalism [4] is based on a postulated internal energy function w for the material,
e
2w Ss 2s
2w 2d
Y
4
It now follows from Eqs. (2)±(4) that for elastic damaging materials: 2 Y 1 s 11 = 2E1 1 2 d1 2 ;
2 Y2 s 22 = 2E2 1 2 d2 2 ;
2 Y12 s 12 = 2G12 1 2 d12 2
5
and we can identify the Yi variables as damage energy release rates in a particular failure mode. The general theory is completed by assuming damage evolution equations in the general form: d1 f1 Y1 ; Y2 ; Y12 ; d12 f12 Y1 ; Y2 ; Y12
d2 f2 Y1 ; Y2 ; Y12 ;
6
where the evolution functions f1, f2, f12 are to be determined. Multi-axial failure, or interaction between damage states, can be included in the model depending on the complexity of the form assumed for these functions. Speci®c forms for the evolution equations are now required for fabric composite plies, which should be consistent with experimental data. For fabric reinforcements it is assumed that there are two main failure mechanisms: ®bre dominated failure in tension or compression in the two ®bre directions; and matrix dominated failure in in-plane shear. Thus, the following simplifying assumptions are made as basis for the elastic damage mechanics fabric model:
3
(a) Fibre and shear damage modes are decoupled, with ®bre damage determined by Y1 and Y2, and shear failure by Y12. (b) Fibre damage development may be different in tension and compression. (c) For balanced fabrics with E1 E2 damage development in the two ®bre directions may be different, thus d1 ± d2 ; but we may assume f1 and f2 have the same functional form so that f1 f2 : (d) The ply material is `nonhealing' thus on unloading after being damaged the damage parameters remain constant until a higher damaging load is re-applied. It follows that the evolution functions depend on the maximum Yi values reached. (e) Damage development does not necessarily lead to ultimate failure of the ply, thus a global failure criterion is also applied.
with the compliance matrix S de®ned above. On introducing a vector of damage parameters d d 1 ; d2 ; d12 T the theory postulates associated `thermodynamic forces' Y Y1 ; Y2 ; Y12 T ; which are `driving forces' for damage
In order to take account of item (d) above the damage evolution equations are based on the maximum value of the damage forces reached during the previous loading history.
w 1=2s T Ss
A.F. Johnson / Composites: Part A 32 (2001) 1197±1206
balanced fabric ply that e11 ; e22 # e1f ; the ply uniaxial failure strain in the ®bre direction.
We thus introduce the quantities Y1, Y2, Y12 de®ned by: p p Y1 t max{ Y1 t}; Y2 t max{ Y2 t}; p t#t Y12 t max{ Y12 t};
7
Note that the pYi quantities are de®ned in terms of the maxima of Y i since test data on UD composites [1] showed that the square root of the damage forces seemed to be the quantity which arises more naturally in the evolution equations. Taking into account (a) and (c) above, and assuming an elastic region without damage at the onset of loading, leading to a lower damage threshold, and cut off at an upper damage threshold, we obtain the speci®c forms for the damage parameters: d1 0; Y1 , Y10 ; d1 a1 Y1 2 Y10 for Y10 , Y1 , Y1f ; otherwise d1 1 d2 0; Y2 , Y10 ; d2 a1 Y2 2 Y10 for Y10 , Y2 , Y1f ; otherwise d2 1
8
d12 0; Y12 , Y120 ; d12 a12 ln Y12 2 ln Y120 for Y120 , Y12 , Y12f ;
1199
otherwise d12 1:
Linear forms have been assumed here for d1 and d2, which were found to be good approximations for UD plies in Ref. [2]. A linear form for d12 in Y12 is unsuitable at larger shear strains in fabric composites, and an equation linear in log (Y12) was found to be a better approximation. Thus the evolution equations for a balanced fabric ply require the determination of two slope parameters a 1, a 12 and four damage threshold parameters Y10, Y120, Y1f, Y12f. In order to take account of different behaviour in tension and compression for ®bre damage (item (b) above), Eq. (8) are generalised by de®ning constants at1 and ac1 ; respectively in the tensile and compressive evolution equations, with corresponding threshold parameters. The tensile values are then used when the volume strain tr e $ 0; and compressive values when tr e , 0: As a further re®nement the damage model is supplemented by additional global failure conditions which override Eq. (8). These de®ne an ultimate failure envelope for the ply, within which properties are degraded according to the damage model, but element stresses are set to zero on reaching the failure envelope even though the damage parameters may not have reached their limiting values. In this way it is possible to bring in recognised multi-axial failure criteria, such as the Tsai-Wu quadratic stress criterion, etc. An example of a global ply failure condition convenient in practice is the maximum ®bre tensile strain condition, so that it is assumed for a
2.2. Elastic±plastic model for shear behaviour The fabric ply model presented above is for an elastic damaging material, which is appropriate for ®bre dominated failure modes such as tensile loading along ®bre directions. For in-plane shear, deformations are controlled by matrix behaviour, which may be inelastic or irreversible due to the presence of extensive matrix cracking or plasticity. On unloading this can lead to permanent deformations in the ply. The extension of the fabric model to include these irreversible damage effects is now considered, based on the additional assumptions: (f) The total strain in the ply is split into the sum of an elastic and inelastic or plastic part. (g) Plastic strains are associated only with matrix dominated in-plane shear response. (h) A classical plasticity model is used with an elastic domain function and hardening law applied to the `effective' stresses in the damaged material. (i) Inelastic or plastic strain increments are assumed to be normal to the elastic domain function. Thus from ( f ) we write the total strain e ee 1 ep as the sum of an elastic e e and plastic strain e p. The elastic strain component is given by Eqs. (2) and (4) as ee Ss: We consider a plane stress model for a thin ply and assume that there are only plastic strains in shear so that: p e 11 ep22 0;
ep12 ± 0
9
Following Ref. [4], an elastic domain function is introduced F s~ 12 ; R where s~ 12 is the `effective' shear stress s~ 12 s 12 = 1 2 d12 and R is an isotropic hardening function. R(p) is a function of an inelastic strain variable p. The elastic domain function F has a simple form here since it is assumed that only the effective shear stress leads to plastic deformation, thus: F s 12 = 1 2 d12 2 R p 2 R0
10
where it is assumed that R 0 0 and that R0 is the initial threshold value for inelastic strain behaviour. The condition F , 0 corresponds to stress states inside the elastic domain where the material is elastic damaging. Inelastic response corresponds to stresses on the elastic domain boundary such that under load increments the stress state remains on the hardening domain boundary. It follows from the normality requirement (i) that F 0; F_ 0; which leads to the condition: 2F e_ p12 l_ 2s 12
2F p_ 2l_ 2R
11
where l_ $ 0 is a proportionality parameter to be determined. On substituting Eq. (10) into Eq. (11) we ®nd that
1200
A.F. Johnson / Composites: Part A 32 (2001) 1197±1206
l_ p_ and hence _ 2 d12 giving p e_ p12 p= 1
Z
ep12 0
1 2 d12 dep12
12
With this de®nition p is the accumulated effective plastic strain over the complete loading cycle. The model is completed by specifying the hardening function R(p). This is determined from cyclic loading tests in which both the elastic and irreversible plastic strains are measured. A typical form assumed for the hardening function [4] is a power law function, and adopting this function here leads to the general equation: R p bpm
13
which depends on the two parameters b and the power index m. This completes the in-plane constitutive model for a fabric composite ply which is elastic damaging in the ®bre directions and in in-plane shear, with elasto±plastic shear response above a shear yield stress R0. The elastic strains are given by Eq. (4), with the required damage parameters d1, d2 and d12 being determined by the damage evolution Eqs. (5) (7) and (8). Plastic strains are obtained by integration of Eq. (12), where p follows from inverting the hardening law (13). In addition to the undamaged ply elastic constants, the model requires experimental determination of the damage evolution constants a 1, a 12, threshold parameters Y10, Y120, Y1f, Y12f and for the inelastic shear contribution the plasticity parameters R0, b and m.
values and with test data on the initial tensile modulus E1. This is shown as the thick line in Fig. 1. Comparison with the measured curve gives good agreement over the complete strain range up to failure, which con®rms the model and the suitability of the assumed linear damage evolution Eq. (8). 3.2. Cyclic shear tests Where irreversible plastic strains occur as in in-plane shear behaviour then a standard shear stress±strain curve under monotonic loads is not suf®cient to determine all the model parameters. In this case a cyclic load test is required in order to measure additionally the permanent plastic shear strain at each stress level, as described in LadeveÁze and Le Dantec [1] for UD plies. A test programme on GF/epoxy under cyclic shear loads was carried out at the DLR, based on tensile tests on GF/epoxy test specimens loaded at 458 to the principal ®bre directions. Fig. 2 is a typical shear stress±strain curve obtained in this way with ®ve load±unload cycles, which shows that there is extensive inelastic or `plastic' deformation in the material. For example in the 4th load cycle the total shear strain e12 0:070 consists of an elastic component ee12 0:038 and plastic component ep12 0:032: The procedure for obtaining the model parameters will now be described, based on the following data recorded for each load cycle. These are the shear stress s 12, the total e p strain e 12, the elastic and plastic strain components e 12 , e 12 and the secant modulus 2Gsec at each cyclic stress level, shown as the dotted lines in Fig. 2. The elastic shear strain component in this test is given from Eqs. (2) and (4) by
3. Determination of fabric ply model parameters
e e 12 s 12 = 2G12 1 2 d12
3.1. Tensile tests along the ®bres
from which it follows that the measured secant shear modulus Gsec is
The parameters of the elastic damaging model are derived from standard stress±strain curves from composites specimen tests loaded monotonically to failure. A strategy has been developed to derive the elastic damage parameters and the plastic hardening law parameters, and is described in detail in Ref. [6]. In the HICAS project mechanical property data for glass fabric/epoxy are reported in Ref. [7]. A typical measured tensile stress±strain curve for loads along the ®bre direction is given in Fig. 1 (thin line) and it deviates from linearity due to microdamage in the matrix and transverse rovings, which reduces the secant modulus at higher strains. From the measured secant modulus we obtain the damage parameter d1 as a function of strain. It rises from zero to a maximum value of about 0.2 before ultimate failure by ®bre fracture. Next the damage evolution equation d1 2 Y1 is plotted, using the de®nitions (5) and (7) to determine Y1. It is found that it can be reasonably well approximated by a straight line as in Eq. (8) and the parameters a 1, Y10 and Y1f required in the model may be determined. As a check on the elastic damage model the theoretical tensile stress±strain curve was computed from the model with these parameters
Gsec s 12 =2ee12 G12 1 2 d12
14
15
Thus the shear damage parameter d12 may be determined from the test data as a function of the applied shear stress and the elastic shear strain. It follows that the shear damage force Y12 may be computed using Eq. (5) and hence the elastic shear damage evolution equation d12 2 Y12 plotted. The resulting evolution curve is shown in Fig. 3 and it is seen that measured damage parameters d12 reach values as high as 0.75. It is apparent that a linear model would only be p valid for small values of Y 12 and would not be a good basis for modelling damage over the complete strain range. A simple function which gives a reasonable ®t to the test data is linear in ln Y 12 and this is also marked as the broken curve in Fig. 3 for comparison. The constants a 12, Y120, and threshold parameter Y12f required in Eq. (8) are readily obtained from the ®tted log function. The plastic hardening function requires the determination of the accumulated plastic strain p from Eq. (12). With the data on the plastic strain ep12 and the damage parameter d12 taken from the cyclic test curve Fig. 2 a plot is made
Stress s11
5.00E-03
1.00E-02
1.50E-02 Strain e11
2.00E-02
2.50E-02
3.00E-02
3.50E-02
Fig. 1. GF/epoxy 08-tensile stress±strain curves: comparison test data with model predictions (test data with acknowledgements to ONERA [7]).
0 0.00E+00
100
200
300
400
500
600
700
800
900
Tensile stress-strain: comparison model and test curve
s s11 - calc
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A.F. Johnson / Composites: Part A 32 (2001) 1197±1206
Shear stress s12 Mpa
Glass fabric/epoxy. cyclic shear stress-strain curve
0
0.02 e12pl
0.04
0.06 e12el
0.08
0.1
0.12
Total shear strain e12
Fig. 2. Measured cyclic shear stress±strain curve for GF/epoxy.
Evolution eqn. - elastic shear 0.9
0.8
0.7
0.6
d12
0.5 d12 Logarithmisch (d12)
0.4
0.3
0.2
0.1
0
-0.1 sq rt Y12
Fig. 3. Elastic damage evolution equation in shear.
A.F. Johnson / Composites: Part A 32 (2001) 1197±1206
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R(p) : comparison test data and power law model 700
600
500
R(p)
400
s12/(1-d12)-Q0 R(p)
300
200
100
0 -0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
-100 p
Fig. 4. Plastic hardening function R(p) in shear.
of 1 2 d12 against plastic strain ep12 and its integration gives the accumulated plastic strain p. The effective shear stress s 12/(1 2 d12) is then plotted against p. On the elastic domain surface it follows from Eq. (10) with F 0 that this gives a graph of R p 1 R 0 against p, from which R0 follows from the intercept at p 0: On subtracting the inelastic threshold stress R0 a graph is obtained of the hardening function R(p) against p, as shown in Fig. 4. This curve is well ®tted by the power law function (13), whence the plasticity parameters b and m are determined. With the undamaged shear modulus G12 we have all the parameters required to characterise the elastic±plastic shear damage model for GF/epoxy. The validity of the model is shown in Fig. 5 by recalculating the cyclic shear stress± strain curve in Fig. 2 using the shear constitutive Eq. (14), with the evolution Eq. (8), the plastic strain relations (12) and (13) and with the model parameters determined here. Fig. 5 indicates a reasonable agreement between the fabric ply model shear stress predictions and the GF/epoxy test data over the complete strain range, and plastic strains were also close to the measured values. At higher strains there is some deviation between re-calculated stresses and original measured values, which is due to several reasons. First the 458-tension test is not valid for determining shear properties at larger strains due to re-orientation of ®bres towards the load direction and hence the presence of ®bre strains, which means that the measured shear stress±strain curve is not accurate at large strains near failure. Torsion tube test would be better but are more expensive to perform. This causes errors in the shear damage and plasticity
parameters used for the model. Second the re-calculated stress±strain curves are very sensitive to these damage parameter values which leads to some overprediction of shear stress in the middle strain range, due to the idealised form of the damage evolution equation, with only three parameters being used to describe a highly nonlinear function. Improved accuracy could therefore be obtained in the model at the expense of more complex forms for the damage evolution curves. 3.3. Plate impact simulation The basic features of the fabric ply damage mechanics model developed here have been implemented as the `Composite global ply model' in the multi-layered MindlinReissner shell element in the commercial crash and impact FE code PAM-CRASH [5]. This includes the complete elastic damage model along both ®bre directions and in in-plane shear, together with plasticity in in-plane shear. An incremental formulation of the model is used with an iterative predictor±corrector algorithm for the calculation of the plastic strains, as discussed further in Ref. [8]. Some validation of the model in PAM-CRASH has been carried out with single elements and test specimen geometries under quasi-static monotonic and cyclic loading using the materials parameters determined here. This gave encouraging results with reasonable agreement between computations and the measured stress±strain curves over most of the strain range. The improved code was then applied to simulate low velocity impact on composite plates. Note
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A.F. Johnson / Composites: Part A 32 (2001) 1197±1206
Shear stress-strain curves with unloading: comparison model and test 180
160
140
Shear stress s12
120
100 S12 (Mpa) s12
80
60
40
20
0 0
0.02
0.04
0.06
0.08
0.1
0.12
To tal strain e12
Fig. 5. GF/epoxy cyclic shear stress±strain curves: comparison test data and model predictions.
that the effect of materials rate dependence is not included in the code version used here, but is being studied in on-going work. Rate dependence of properties is important for GF/ epoxy composites, and for CF/epoxy it is signi®cant in inplane shear. However shear failures are not the dominant failure mode in the plate transverse impact tests here. In the impact test setup composite plates are simply supported on a square steel frame in the DLR drop tower load platform, and are impacted at the plate centre. The plates were fabricated from 16 plies of carbon fabric/ epoxy prepreg material with a quasi-isotropic layup and a nominal thickness of 4.5 mm. They were 300 £ 300 mm 2 and simply supported on a rigid frame of 250 £ 250 mm 2. The impactor head was a 50 mm diameter steel hemisphere with an added mass of 21 kg and a test series was carried out with impact velocities in the range 2.33±6.28 m/s to give a range of different failure modes from rebound to full penetration. The impactor head was instrumented with a load cell so that the load±time response during impact could be measured. In the 6.28 m/s impact test with impact energy of 414 J the impactor penetrated the plate causing signi®cant in-plane damage and ®bre failure. The plate was modelled by layered shell elements with quasi-isotropic ply layup, placed in contact with the rigid support frame and impacted at the centre by a rigid hemispherical impactor with mass 21 kg and impact velocity 6.28 m/s. The appropriate parameters of the ply damage model were determined for CF/epoxy based on an in-plane
materials test programme similar to that described above for GF/epoxy. Fig. 6 shows the predicted plate response at 3.2 ms as the impactor penetrates the plate causing ply damage and ®bre fracture. This agrees well with the impact damage observed in the corresponding test. An important feature of the implemented fabric model is that it distinguishes clearly between different failure modes in the structure. In post-processing it is possible to follow the progression of the ®bre and shear damage parameters, plastic shear strains, and total strains. As an example Fig. 6 plots contours of maximum ply damage in the layered shell elements, showing how in-plane damage is localised near the impactor. A quantitative assessment of the impact simulation is obtained by comparing measured and predicted contact forces. In the test a quartz load cell is placed behind the hemi-spherical impactor head and the measured load±time response during impact is plotted in Fig. 7. This is compared in the ®gure with the vertical component of the simulated contact force between the impactor head and the plate. The simulated and measured load pulse shape and response time are seen to be in reasonable agreement, but the simulated load amplitudes are signi®cantly higher than those measured. This result is typical of previous experience with plate impact simulations and is often explained by the poor modelling of the plate ®xing conditions. It is known that peak loads are strongly dependent on plate boundary conditions with clamped edges leading to much
A.F. Johnson / Composites: Part A 32 (2001) 1197±1206
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Fig. 6. CF/epoxy plate impact simulation after 3.2 ms, showing penetration by the impactor and contours of ply damage V0 6:28 m=s; M 21 kg:
12,0 Experiment Simulation 10,0
Load [kN]
8,0
6,0
4,0
2,0
0,0 0
1
2
3
4
5
6
7
8
Time [ms] Fig. 7. CF/epoxy plate impact simulation: comparison of computed and measured impactor load pulse V0 6:28 m=s; M 21 kg:
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higher predicted loads than simply supported edges, neither of which can be achieved exactly in tests. In the tests here, the plates were placed on a supporting frame and lightly fastened only at the centre points of the edges, so that the assumption of simple supports should be a reasonable one. The reason for the discrepancy between measured and predicted peak loads is therefore more likely to be explained by the neglect of delamination in the composites model. Although the ply damage was localised near the impactor, C-scans on damaged plates showed a much larger delaminated region with maximum diameter about 120 mm. In this region, the plate ¯exural rigidity could be considerably lower than in a perfectly bonded laminate, depending on the number of delaminations through the thickness. It is expected that this reduced ¯exural stiffness will lower measured peak loads during impact, and could explain the discrepancy between simulation and test results seen here. In ongoing research a composites delamination model has been developed and implemented into PAM-CRASH, and a more careful analysis of the plate impact tests has been carried out [8] which support the importance of modelling both ply damage and delamination during impact. 4. Concluding remarks The paper discusses a damage mechanics model for composite shell elements with ®bre fabric reinforcement, as the ®rst part of a composites model suitable for implementation into explicit FE codes for predicting damage and failure during impact of composite structures. The model developed contains elastic damage along the principal ®bre directions, with an elastic±plastic damaging behaviour in in-plane shear. Damage evolution equations are assumed to take simple linear or log linear forms, and a power law function is assumed for the plastic hardening function. It is shown how the model parameters can be determined from a basic materials test programme. This composites fabric ply model has been implemented in the explicit FE code PAMCRASH, and has been tested on single elements under monotonic and cyclic loads. A low velocity plate impact simulation, carried out with the code to demonstrate the
model, gave encouraging results on failure mode prediction but showed up the model limitations under low velocity impact where delamination damage may be signi®cant. Further developments in composites modelling under consideration include methods for bringing composites rate dependent properties into FE codes and investigation of new numerical approaches for modelling composites delamination. The delamination model to be reported in Ref. [8], is based on modelling a composites laminate by stacked shell elements which have degraded interface stiffnesses as delaminations develop. Further validation of these code improvements for impact simulations based on a gas gun impact test programme on composite structures has been carried out in the CEC supported project HICAS [2]. Acknowledgements The work presented here was developed in the EU project HICAS [2]. The author wishes to acknowledge the ®nancial contribution of the CEC, and the HICAS partners for their contribution to the materials test programme and modelling developments, in particular Dr A.K. Pickett (ESI) for the PAM-CRASH developments and Mr D. Delsart (ONERA) for the materials test data in Fig. 1. References [1] LadeveÁze P, Le Dantec E. Damage modelling of the elementary ply for laminated composites. Compos Sci Technol 1992;43:257±67. [2] High velocity impact of composite aircraft structures (HICAS), CEC DG XII BRITE-EURAM Project BE 96-4238, 1998. [3] Talreja R, editor. Damage mechanics of composite materials. Composite Materials Series, vol. 9. 1994. [4] LadeveÁze P. Inelastic strains and damage. In: Talreja R, editor. Damage mechanics of composite materials. Composite Materials Series, vol. 9, Amsterdam: Elsevier, 1994, Chapt. 4. [5] PAM-CRASHe FE Code, Engineering Systems International, 20 Rue Saarinen, Silic 270, 94578 Rungis-Cedex, France. [6] Johnson AF. In-plane damage models for fabric reinforced composites. HICAS D.2.1.2, DLR Report: DLR-IB 435-99/24, Stuttgart, 1999. [7] Delsart D. HR test data on material N0. 3, HICAS D.1.3.3, ONERAIMFL Report 98/36, Lille, 1998. [8] Johnson AF, Pickett AK, Rozycki P. In preparation.
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Modelling fabric reinforced composites under impact loads A.F. Johnson* German Aerospace Center (DLR), Institute of Structures and Design, Pfaffenwaldring 38-40, D-70569 Stuttgart, Germany Received 21 March 2000; revised 6 November 2000; accepted 4 December 2000
Abstract The paper describes recent progress on the materials modelling and numerical simulation of the in-plane response of ®bre reinforced composite structures. A continuum damage mechanics model for fabric reinforced composites under in-plane loads is presented. It is based on methods developed for UD ply materials (Compos. Sci. Technol., 43 (1992) 257), which are generalised here to fabric reinforcements. The model contains elastic damage in the ®bre directions, with an elastic±plastic model for inelastic shear effects. Test data on a glass fabric/ epoxy laminate show the importance of inelastic effects in shear. A strategy is described for determining model parameters from the test data. The fabric model is being implemented in an explicit FE code for use in crash and impact studies and preliminary results are presented on a plate impact simulation. q 2001 Published by Elsevier Science Ltd. Keywords: Composite materials; C. Damage mechanics
1. Introduction Composite materials are now being used in primary aircraft structures, particularly in helicopters, light aircraft, commuter planes and sailplanes, because of numerous advantages including low weight, high static and fatigue strength and the possibility to manufacture large integral shell structures. Materials such as carbon ®bre/epoxy are inherently brittle and usually exhibit a linear elastic response up to failure with little or no plasticity. Thus composite structures are vulnerable to impact damage and have to satisfy certi®cation procedures for high velocity impact from runway debris or bird strike. Conventional metallic structures absorb impact and crash energy through plastic deformation and folding. Modern explicit FE codes are able to model these effects and are being successfully applied to simulate the collapse of metallic aircraft and automotive structures. This paper is concerned with the development and validity of such codes for modelling the response of composite structures under impact loads. This topic is being studied in some detail within a CEC funded research project on `High velocity impact of composite aircraft structures' HICAS [2]. This project includes an extensive composites materials and
* Tel.: 149-711-686-2297; fax: 149-711-686-2227. E-mail address: [email protected] (A.F. Johnson). 1359-835X/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII: S 1359-835 X(00)00 186-X
structures test programme, composites modelling developments, FE code implementation and impact simulations. Two important aspects of impact modelling are delamination, which is important in lower energy impacts and in failure initiation, and in-plane ply failure, which controls ultimate failure and penetration in the structure. This paper summarises some of the modelling developments being carried out at the DLR within the project on in-plane damage models for fabric reinforced composite plies, since fabric reinforcements are of considerable practical interest for aircraft structures. Related work is considering delamination failure models and rate dependent behaviour. Emphasis is given to composite materials models suitable for implementation into FE codes, which can adequately characterise the nonlinear damage progression and different failure modes that occur in fabric plies. A continuum damage mechanics model for fabric reinforced composites under in-plane loads is presented. It is based on methods developed for UD ply materials by LadeveÁze and co-workers in Refs. [1,4], which are generalised here to fabric reinforcements. The model contains elastic damage in the ®bre directions, with an elastic±plastic model for inelastic shear effects. A strategy is then described for determining model parameters from the test data. The fabric model is being implemented into PAMCRASH [5], a commercial explicit FE code, and preliminary results are presented on the simulation of ply damage and failure in plate impact.
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development. It can then be shown that the strains e and the damage forces Y are de®ned from the strain energy function by:
2. Composites fabric ply damage model 2.1. Elastic damage mechanics model The fabric reinforced composite ply is modelled as a homogeneous orthotropic elastic or elastic±plastic damaging material whose properties are degraded on loading by microcracking prior to ultimate failure. A continuum damage mechanics formulation is used [3] in which ply degradation parameters are internal state variables, which are governed by damage evolution equations. General constitutive laws for orthotropic elastic materials with internal damage parameters are described by LadeveÁze [4]. In the plane stress case required here to characterise the properties of composite plies or shell elements with orthotropic symmetry axes (x1, x2), the in-plane stress and strain components are
s s 11 ; s 22 ; s 12 T
e e11 ; e22 ; 2e12 T :
1
Using a strain equivalent damage mechanics formulation, an elastic compliance matrix S is assumed with the general form: 0 1 1=E1 1 2 d1 2n12 =E1 0 B C C 2 SB 1=E2 1 2 d2 0 @ 2n12 =E1 A 0
0
1=G12 1 2 d12
The compliance matrix has three scalar damage parameters d1, d2, d12 and four initial or `undamaged' elastic constants: Young's moduli in the principal orthotropy directions E1, E2, the in-plane shear modulus G12, and the principal Poisson's ratio n 12. The damage parameters have values 0 # di # 1 and represent modulus reductions under different loading conditions due to microdamage in the material. Thus for fabric plies d1 and d2 are associated with damage or failure in the principal ®bre directions, and d12 with in-plane shear failure. In general, they will be invariant functions of suitable physical parameters, which characterise the damage state. In the present model, for simplicity, there is no additional damage parameter to model independent degradation in Poisson's ratios. The effect of this assumption is that as damage progresses n 12 is reduced by the factor 1 2 d1 ; since for a uniaxial stress ®eld s 11 it can be shown from Eqs. (1) and (2) that 2e 22 =e11 n12 1 2 d1 ; Similarly it is found under pure transverse stress s 22 that n 21 is reduced by the factor 1 2 d 2 : These assumptions are considered reasonable for the ®rst order damage evolution model described in this paper. The general damage mechanics formalism [4] is based on a postulated internal energy function w for the material,
e
2w Ss 2s
2w 2d
Y
4
It now follows from Eqs. (2)±(4) that for elastic damaging materials: 2 Y 1 s 11 = 2E1 1 2 d1 2 ;
2 Y2 s 22 = 2E2 1 2 d2 2 ;
2 Y12 s 12 = 2G12 1 2 d12 2
5
and we can identify the Yi variables as damage energy release rates in a particular failure mode. The general theory is completed by assuming damage evolution equations in the general form: d1 f1 Y1 ; Y2 ; Y12 ; d12 f12 Y1 ; Y2 ; Y12
d2 f2 Y1 ; Y2 ; Y12 ;
6
where the evolution functions f1, f2, f12 are to be determined. Multi-axial failure, or interaction between damage states, can be included in the model depending on the complexity of the form assumed for these functions. Speci®c forms for the evolution equations are now required for fabric composite plies, which should be consistent with experimental data. For fabric reinforcements it is assumed that there are two main failure mechanisms: ®bre dominated failure in tension or compression in the two ®bre directions; and matrix dominated failure in in-plane shear. Thus, the following simplifying assumptions are made as basis for the elastic damage mechanics fabric model:
3
(a) Fibre and shear damage modes are decoupled, with ®bre damage determined by Y1 and Y2, and shear failure by Y12. (b) Fibre damage development may be different in tension and compression. (c) For balanced fabrics with E1 E2 damage development in the two ®bre directions may be different, thus d1 ± d2 ; but we may assume f1 and f2 have the same functional form so that f1 f2 : (d) The ply material is `nonhealing' thus on unloading after being damaged the damage parameters remain constant until a higher damaging load is re-applied. It follows that the evolution functions depend on the maximum Yi values reached. (e) Damage development does not necessarily lead to ultimate failure of the ply, thus a global failure criterion is also applied.
with the compliance matrix S de®ned above. On introducing a vector of damage parameters d d 1 ; d2 ; d12 T the theory postulates associated `thermodynamic forces' Y Y1 ; Y2 ; Y12 T ; which are `driving forces' for damage
In order to take account of item (d) above the damage evolution equations are based on the maximum value of the damage forces reached during the previous loading history.
w 1=2s T Ss
A.F. Johnson / Composites: Part A 32 (2001) 1197±1206
balanced fabric ply that e11 ; e22 # e1f ; the ply uniaxial failure strain in the ®bre direction.
We thus introduce the quantities Y1, Y2, Y12 de®ned by: p p Y1 t max{ Y1 t}; Y2 t max{ Y2 t}; p t#t Y12 t max{ Y12 t};
7
Note that the pYi quantities are de®ned in terms of the maxima of Y i since test data on UD composites [1] showed that the square root of the damage forces seemed to be the quantity which arises more naturally in the evolution equations. Taking into account (a) and (c) above, and assuming an elastic region without damage at the onset of loading, leading to a lower damage threshold, and cut off at an upper damage threshold, we obtain the speci®c forms for the damage parameters: d1 0; Y1 , Y10 ; d1 a1 Y1 2 Y10 for Y10 , Y1 , Y1f ; otherwise d1 1 d2 0; Y2 , Y10 ; d2 a1 Y2 2 Y10 for Y10 , Y2 , Y1f ; otherwise d2 1
8
d12 0; Y12 , Y120 ; d12 a12 ln Y12 2 ln Y120 for Y120 , Y12 , Y12f ;
1199
otherwise d12 1:
Linear forms have been assumed here for d1 and d2, which were found to be good approximations for UD plies in Ref. [2]. A linear form for d12 in Y12 is unsuitable at larger shear strains in fabric composites, and an equation linear in log (Y12) was found to be a better approximation. Thus the evolution equations for a balanced fabric ply require the determination of two slope parameters a 1, a 12 and four damage threshold parameters Y10, Y120, Y1f, Y12f. In order to take account of different behaviour in tension and compression for ®bre damage (item (b) above), Eq. (8) are generalised by de®ning constants at1 and ac1 ; respectively in the tensile and compressive evolution equations, with corresponding threshold parameters. The tensile values are then used when the volume strain tr e $ 0; and compressive values when tr e , 0: As a further re®nement the damage model is supplemented by additional global failure conditions which override Eq. (8). These de®ne an ultimate failure envelope for the ply, within which properties are degraded according to the damage model, but element stresses are set to zero on reaching the failure envelope even though the damage parameters may not have reached their limiting values. In this way it is possible to bring in recognised multi-axial failure criteria, such as the Tsai-Wu quadratic stress criterion, etc. An example of a global ply failure condition convenient in practice is the maximum ®bre tensile strain condition, so that it is assumed for a
2.2. Elastic±plastic model for shear behaviour The fabric ply model presented above is for an elastic damaging material, which is appropriate for ®bre dominated failure modes such as tensile loading along ®bre directions. For in-plane shear, deformations are controlled by matrix behaviour, which may be inelastic or irreversible due to the presence of extensive matrix cracking or plasticity. On unloading this can lead to permanent deformations in the ply. The extension of the fabric model to include these irreversible damage effects is now considered, based on the additional assumptions: (f) The total strain in the ply is split into the sum of an elastic and inelastic or plastic part. (g) Plastic strains are associated only with matrix dominated in-plane shear response. (h) A classical plasticity model is used with an elastic domain function and hardening law applied to the `effective' stresses in the damaged material. (i) Inelastic or plastic strain increments are assumed to be normal to the elastic domain function. Thus from ( f ) we write the total strain e ee 1 ep as the sum of an elastic e e and plastic strain e p. The elastic strain component is given by Eqs. (2) and (4) as ee Ss: We consider a plane stress model for a thin ply and assume that there are only plastic strains in shear so that: p e 11 ep22 0;
ep12 ± 0
9
Following Ref. [4], an elastic domain function is introduced F s~ 12 ; R where s~ 12 is the `effective' shear stress s~ 12 s 12 = 1 2 d12 and R is an isotropic hardening function. R(p) is a function of an inelastic strain variable p. The elastic domain function F has a simple form here since it is assumed that only the effective shear stress leads to plastic deformation, thus: F s 12 = 1 2 d12 2 R p 2 R0
10
where it is assumed that R 0 0 and that R0 is the initial threshold value for inelastic strain behaviour. The condition F , 0 corresponds to stress states inside the elastic domain where the material is elastic damaging. Inelastic response corresponds to stresses on the elastic domain boundary such that under load increments the stress state remains on the hardening domain boundary. It follows from the normality requirement (i) that F 0; F_ 0; which leads to the condition: 2F e_ p12 l_ 2s 12
2F p_ 2l_ 2R
11
where l_ $ 0 is a proportionality parameter to be determined. On substituting Eq. (10) into Eq. (11) we ®nd that
1200
A.F. Johnson / Composites: Part A 32 (2001) 1197±1206
l_ p_ and hence _ 2 d12 giving p e_ p12 p= 1
Z
ep12 0
1 2 d12 dep12
12
With this de®nition p is the accumulated effective plastic strain over the complete loading cycle. The model is completed by specifying the hardening function R(p). This is determined from cyclic loading tests in which both the elastic and irreversible plastic strains are measured. A typical form assumed for the hardening function [4] is a power law function, and adopting this function here leads to the general equation: R p bpm
13
which depends on the two parameters b and the power index m. This completes the in-plane constitutive model for a fabric composite ply which is elastic damaging in the ®bre directions and in in-plane shear, with elasto±plastic shear response above a shear yield stress R0. The elastic strains are given by Eq. (4), with the required damage parameters d1, d2 and d12 being determined by the damage evolution Eqs. (5) (7) and (8). Plastic strains are obtained by integration of Eq. (12), where p follows from inverting the hardening law (13). In addition to the undamaged ply elastic constants, the model requires experimental determination of the damage evolution constants a 1, a 12, threshold parameters Y10, Y120, Y1f, Y12f and for the inelastic shear contribution the plasticity parameters R0, b and m.
values and with test data on the initial tensile modulus E1. This is shown as the thick line in Fig. 1. Comparison with the measured curve gives good agreement over the complete strain range up to failure, which con®rms the model and the suitability of the assumed linear damage evolution Eq. (8). 3.2. Cyclic shear tests Where irreversible plastic strains occur as in in-plane shear behaviour then a standard shear stress±strain curve under monotonic loads is not suf®cient to determine all the model parameters. In this case a cyclic load test is required in order to measure additionally the permanent plastic shear strain at each stress level, as described in LadeveÁze and Le Dantec [1] for UD plies. A test programme on GF/epoxy under cyclic shear loads was carried out at the DLR, based on tensile tests on GF/epoxy test specimens loaded at 458 to the principal ®bre directions. Fig. 2 is a typical shear stress±strain curve obtained in this way with ®ve load±unload cycles, which shows that there is extensive inelastic or `plastic' deformation in the material. For example in the 4th load cycle the total shear strain e12 0:070 consists of an elastic component ee12 0:038 and plastic component ep12 0:032: The procedure for obtaining the model parameters will now be described, based on the following data recorded for each load cycle. These are the shear stress s 12, the total e p strain e 12, the elastic and plastic strain components e 12 , e 12 and the secant modulus 2Gsec at each cyclic stress level, shown as the dotted lines in Fig. 2. The elastic shear strain component in this test is given from Eqs. (2) and (4) by
3. Determination of fabric ply model parameters
e e 12 s 12 = 2G12 1 2 d12
3.1. Tensile tests along the ®bres
from which it follows that the measured secant shear modulus Gsec is
The parameters of the elastic damaging model are derived from standard stress±strain curves from composites specimen tests loaded monotonically to failure. A strategy has been developed to derive the elastic damage parameters and the plastic hardening law parameters, and is described in detail in Ref. [6]. In the HICAS project mechanical property data for glass fabric/epoxy are reported in Ref. [7]. A typical measured tensile stress±strain curve for loads along the ®bre direction is given in Fig. 1 (thin line) and it deviates from linearity due to microdamage in the matrix and transverse rovings, which reduces the secant modulus at higher strains. From the measured secant modulus we obtain the damage parameter d1 as a function of strain. It rises from zero to a maximum value of about 0.2 before ultimate failure by ®bre fracture. Next the damage evolution equation d1 2 Y1 is plotted, using the de®nitions (5) and (7) to determine Y1. It is found that it can be reasonably well approximated by a straight line as in Eq. (8) and the parameters a 1, Y10 and Y1f required in the model may be determined. As a check on the elastic damage model the theoretical tensile stress±strain curve was computed from the model with these parameters
Gsec s 12 =2ee12 G12 1 2 d12
14
15
Thus the shear damage parameter d12 may be determined from the test data as a function of the applied shear stress and the elastic shear strain. It follows that the shear damage force Y12 may be computed using Eq. (5) and hence the elastic shear damage evolution equation d12 2 Y12 plotted. The resulting evolution curve is shown in Fig. 3 and it is seen that measured damage parameters d12 reach values as high as 0.75. It is apparent that a linear model would only be p valid for small values of Y 12 and would not be a good basis for modelling damage over the complete strain range. A simple function which gives a reasonable ®t to the test data is linear in ln Y 12 and this is also marked as the broken curve in Fig. 3 for comparison. The constants a 12, Y120, and threshold parameter Y12f required in Eq. (8) are readily obtained from the ®tted log function. The plastic hardening function requires the determination of the accumulated plastic strain p from Eq. (12). With the data on the plastic strain ep12 and the damage parameter d12 taken from the cyclic test curve Fig. 2 a plot is made
Stress s11
5.00E-03
1.00E-02
1.50E-02 Strain e11
2.00E-02
2.50E-02
3.00E-02
3.50E-02
Fig. 1. GF/epoxy 08-tensile stress±strain curves: comparison test data with model predictions (test data with acknowledgements to ONERA [7]).
0 0.00E+00
100
200
300
400
500
600
700
800
900
Tensile stress-strain: comparison model and test curve
s s11 - calc
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A.F. Johnson / Composites: Part A 32 (2001) 1197±1206
Shear stress s12 Mpa
Glass fabric/epoxy. cyclic shear stress-strain curve
0
0.02 e12pl
0.04
0.06 e12el
0.08
0.1
0.12
Total shear strain e12
Fig. 2. Measured cyclic shear stress±strain curve for GF/epoxy.
Evolution eqn. - elastic shear 0.9
0.8
0.7
0.6
d12
0.5 d12 Logarithmisch (d12)
0.4
0.3
0.2
0.1
0
-0.1 sq rt Y12
Fig. 3. Elastic damage evolution equation in shear.
A.F. Johnson / Composites: Part A 32 (2001) 1197±1206
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R(p) : comparison test data and power law model 700
600
500
R(p)
400
s12/(1-d12)-Q0 R(p)
300
200
100
0 -0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
-100 p
Fig. 4. Plastic hardening function R(p) in shear.
of 1 2 d12 against plastic strain ep12 and its integration gives the accumulated plastic strain p. The effective shear stress s 12/(1 2 d12) is then plotted against p. On the elastic domain surface it follows from Eq. (10) with F 0 that this gives a graph of R p 1 R 0 against p, from which R0 follows from the intercept at p 0: On subtracting the inelastic threshold stress R0 a graph is obtained of the hardening function R(p) against p, as shown in Fig. 4. This curve is well ®tted by the power law function (13), whence the plasticity parameters b and m are determined. With the undamaged shear modulus G12 we have all the parameters required to characterise the elastic±plastic shear damage model for GF/epoxy. The validity of the model is shown in Fig. 5 by recalculating the cyclic shear stress± strain curve in Fig. 2 using the shear constitutive Eq. (14), with the evolution Eq. (8), the plastic strain relations (12) and (13) and with the model parameters determined here. Fig. 5 indicates a reasonable agreement between the fabric ply model shear stress predictions and the GF/epoxy test data over the complete strain range, and plastic strains were also close to the measured values. At higher strains there is some deviation between re-calculated stresses and original measured values, which is due to several reasons. First the 458-tension test is not valid for determining shear properties at larger strains due to re-orientation of ®bres towards the load direction and hence the presence of ®bre strains, which means that the measured shear stress±strain curve is not accurate at large strains near failure. Torsion tube test would be better but are more expensive to perform. This causes errors in the shear damage and plasticity
parameters used for the model. Second the re-calculated stress±strain curves are very sensitive to these damage parameter values which leads to some overprediction of shear stress in the middle strain range, due to the idealised form of the damage evolution equation, with only three parameters being used to describe a highly nonlinear function. Improved accuracy could therefore be obtained in the model at the expense of more complex forms for the damage evolution curves. 3.3. Plate impact simulation The basic features of the fabric ply damage mechanics model developed here have been implemented as the `Composite global ply model' in the multi-layered MindlinReissner shell element in the commercial crash and impact FE code PAM-CRASH [5]. This includes the complete elastic damage model along both ®bre directions and in in-plane shear, together with plasticity in in-plane shear. An incremental formulation of the model is used with an iterative predictor±corrector algorithm for the calculation of the plastic strains, as discussed further in Ref. [8]. Some validation of the model in PAM-CRASH has been carried out with single elements and test specimen geometries under quasi-static monotonic and cyclic loading using the materials parameters determined here. This gave encouraging results with reasonable agreement between computations and the measured stress±strain curves over most of the strain range. The improved code was then applied to simulate low velocity impact on composite plates. Note
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Shear stress-strain curves with unloading: comparison model and test 180
160
140
Shear stress s12
120
100 S12 (Mpa) s12
80
60
40
20
0 0
0.02
0.04
0.06
0.08
0.1
0.12
To tal strain e12
Fig. 5. GF/epoxy cyclic shear stress±strain curves: comparison test data and model predictions.
that the effect of materials rate dependence is not included in the code version used here, but is being studied in on-going work. Rate dependence of properties is important for GF/ epoxy composites, and for CF/epoxy it is signi®cant in inplane shear. However shear failures are not the dominant failure mode in the plate transverse impact tests here. In the impact test setup composite plates are simply supported on a square steel frame in the DLR drop tower load platform, and are impacted at the plate centre. The plates were fabricated from 16 plies of carbon fabric/ epoxy prepreg material with a quasi-isotropic layup and a nominal thickness of 4.5 mm. They were 300 £ 300 mm 2 and simply supported on a rigid frame of 250 £ 250 mm 2. The impactor head was a 50 mm diameter steel hemisphere with an added mass of 21 kg and a test series was carried out with impact velocities in the range 2.33±6.28 m/s to give a range of different failure modes from rebound to full penetration. The impactor head was instrumented with a load cell so that the load±time response during impact could be measured. In the 6.28 m/s impact test with impact energy of 414 J the impactor penetrated the plate causing signi®cant in-plane damage and ®bre failure. The plate was modelled by layered shell elements with quasi-isotropic ply layup, placed in contact with the rigid support frame and impacted at the centre by a rigid hemispherical impactor with mass 21 kg and impact velocity 6.28 m/s. The appropriate parameters of the ply damage model were determined for CF/epoxy based on an in-plane
materials test programme similar to that described above for GF/epoxy. Fig. 6 shows the predicted plate response at 3.2 ms as the impactor penetrates the plate causing ply damage and ®bre fracture. This agrees well with the impact damage observed in the corresponding test. An important feature of the implemented fabric model is that it distinguishes clearly between different failure modes in the structure. In post-processing it is possible to follow the progression of the ®bre and shear damage parameters, plastic shear strains, and total strains. As an example Fig. 6 plots contours of maximum ply damage in the layered shell elements, showing how in-plane damage is localised near the impactor. A quantitative assessment of the impact simulation is obtained by comparing measured and predicted contact forces. In the test a quartz load cell is placed behind the hemi-spherical impactor head and the measured load±time response during impact is plotted in Fig. 7. This is compared in the ®gure with the vertical component of the simulated contact force between the impactor head and the plate. The simulated and measured load pulse shape and response time are seen to be in reasonable agreement, but the simulated load amplitudes are signi®cantly higher than those measured. This result is typical of previous experience with plate impact simulations and is often explained by the poor modelling of the plate ®xing conditions. It is known that peak loads are strongly dependent on plate boundary conditions with clamped edges leading to much
A.F. Johnson / Composites: Part A 32 (2001) 1197±1206
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Fig. 6. CF/epoxy plate impact simulation after 3.2 ms, showing penetration by the impactor and contours of ply damage V0 6:28 m=s; M 21 kg:
12,0 Experiment Simulation 10,0
Load [kN]
8,0
6,0
4,0
2,0
0,0 0
1
2
3
4
5
6
7
8
Time [ms] Fig. 7. CF/epoxy plate impact simulation: comparison of computed and measured impactor load pulse V0 6:28 m=s; M 21 kg:
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higher predicted loads than simply supported edges, neither of which can be achieved exactly in tests. In the tests here, the plates were placed on a supporting frame and lightly fastened only at the centre points of the edges, so that the assumption of simple supports should be a reasonable one. The reason for the discrepancy between measured and predicted peak loads is therefore more likely to be explained by the neglect of delamination in the composites model. Although the ply damage was localised near the impactor, C-scans on damaged plates showed a much larger delaminated region with maximum diameter about 120 mm. In this region, the plate ¯exural rigidity could be considerably lower than in a perfectly bonded laminate, depending on the number of delaminations through the thickness. It is expected that this reduced ¯exural stiffness will lower measured peak loads during impact, and could explain the discrepancy between simulation and test results seen here. In ongoing research a composites delamination model has been developed and implemented into PAM-CRASH, and a more careful analysis of the plate impact tests has been carried out [8] which support the importance of modelling both ply damage and delamination during impact. 4. Concluding remarks The paper discusses a damage mechanics model for composite shell elements with ®bre fabric reinforcement, as the ®rst part of a composites model suitable for implementation into explicit FE codes for predicting damage and failure during impact of composite structures. The model developed contains elastic damage along the principal ®bre directions, with an elastic±plastic damaging behaviour in in-plane shear. Damage evolution equations are assumed to take simple linear or log linear forms, and a power law function is assumed for the plastic hardening function. It is shown how the model parameters can be determined from a basic materials test programme. This composites fabric ply model has been implemented in the explicit FE code PAMCRASH, and has been tested on single elements under monotonic and cyclic loads. A low velocity plate impact simulation, carried out with the code to demonstrate the
model, gave encouraging results on failure mode prediction but showed up the model limitations under low velocity impact where delamination damage may be signi®cant. Further developments in composites modelling under consideration include methods for bringing composites rate dependent properties into FE codes and investigation of new numerical approaches for modelling composites delamination. The delamination model to be reported in Ref. [8], is based on modelling a composites laminate by stacked shell elements which have degraded interface stiffnesses as delaminations develop. Further validation of these code improvements for impact simulations based on a gas gun impact test programme on composite structures has been carried out in the CEC supported project HICAS [2]. Acknowledgements The work presented here was developed in the EU project HICAS [2]. The author wishes to acknowledge the ®nancial contribution of the CEC, and the HICAS partners for their contribution to the materials test programme and modelling developments, in particular Dr A.K. Pickett (ESI) for the PAM-CRASH developments and Mr D. Delsart (ONERA) for the materials test data in Fig. 1. References [1] LadeveÁze P, Le Dantec E. Damage modelling of the elementary ply for laminated composites. Compos Sci Technol 1992;43:257±67. [2] High velocity impact of composite aircraft structures (HICAS), CEC DG XII BRITE-EURAM Project BE 96-4238, 1998. [3] Talreja R, editor. Damage mechanics of composite materials. Composite Materials Series, vol. 9. 1994. [4] LadeveÁze P. Inelastic strains and damage. In: Talreja R, editor. Damage mechanics of composite materials. Composite Materials Series, vol. 9, Amsterdam: Elsevier, 1994, Chapt. 4. [5] PAM-CRASHe FE Code, Engineering Systems International, 20 Rue Saarinen, Silic 270, 94578 Rungis-Cedex, France. [6] Johnson AF. In-plane damage models for fabric reinforced composites. HICAS D.2.1.2, DLR Report: DLR-IB 435-99/24, Stuttgart, 1999. [7] Delsart D. HR test data on material N0. 3, HICAS D.1.3.3, ONERAIMFL Report 98/36, Lille, 1998. [8] Johnson AF, Pickett AK, Rozycki P. In preparation.