RATE-DEPENDENT MECHANICAL BEHAVIOR OF CFRP LAMINATES UNDER COMPRESSION LOADING

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1 RATE-DEPENDENT MECHANICAL BEHAVIOR OF CFRP LAMINATES UNDER COMPRESSION LOADING R. M. Guedes 1,, P. H. Magalhães, F. Ferreira 3 and J. L. Morais 4 1 Deartamento de Engenharia Mecânica e Gestão Industrial, Universidade do Porto, Rua Dr. Roberto Frias s/n, Porto, Portugal Instituto de Engenharia Mecânica e Gestão Industrial, Rua do Barroco, 174, Leça do Balio, Portugal 3 Instituto Suerior de Engenharia do Porto, Rua António Bernardino de Almeida 431, 4-7 Porto, Portugal 3 Deartamento de Engenharias, Universidade de Trás-os-Montes e Alto Douro, Quinta de Prados, Vila Real, Portugal ABSTRACT This work describes an attemt to characterize the rate-deendent non-linear behaviour of a olymeric comosite material under monotonic comression loading in the range of.1 to 4/s. A 3-arameter constitutive viscolastic model [1,] is used to describe the mechanical behaviour. This model was develoed based on data for strain rate between.1 and.7/s. In transverse direction the viscolastic model was able to redict the high strain rate eeriments conducted on a Slit Hokinson Pressure Bar. 1. INTRODUCTION The influence of strain rate on comression roerties of a CFRP is investigated and resented. This is art of a major research roject aiming to study the mechanical behavior and failure of CFRP laminates under high strain rate. The increasing use of olymer matri comosite materials in many alications demands a characterization of their mechanical resonse under a range of strain rates from 1-3 to 1 +3 /s. Several eerimental studies have been erformed with the goal of determining the effects of strain rate on the mechanical roerties and resonse of olymer matri comosite systems at high strain rate conditions. A survey done by Gates [3] describes several analytical models to simulate the rate-deendent resonse of various tyes of olymer matri comosites. Later Gates and Sun [3] develoed an elastic-viscolastic model for an orthotroic material without tension/comression symmetry and Gates [4] used an etension of that model to describe an eerimental methodology to generate material constants. Goldberg and Stouffer [5,6] resented an historical descrition of eerimental develoment to study the strain rate effect on mechanical roerties of olymer matri comosites, followed by constitutive modeling. These authors modified strain rate deendent inelastic constitutive models, used to model viscolastic deformation of metals, to model non-linear viscoelastic deformation of olymers. It is reasonably acceted that in the fiber direction, unidirectional fiber reinforced olymer matri comosites are insensitive to load level and time of load alication, i.e. ehibit a linear behavior. Then again in all other directions olymer comosites ehibit non-linear and time-deendent behavior, which also means a rate-deendent behavior. A quit simle methodology associated to a simle constitutive elastic-viscolastic model was develoed by Sun et al.[1-]. This methodology was alied in the resent work to study mechanical behavior of a comosite laminate in the transverse direction with good success In the resent work the material system used was Teireg HS16 REM, comrising high strength carbon fiber and eoy resin. The average thickness of cured laminates was.5 mm. Square samles with 8.5 mm side were cut using water jet from original laminates. The unidirectional couon secimens were subjected to loading along the 3º, 45º, 75º and 9º directions.

2 . THEORETICAL BACKGROUND The 3-arameter viscolastic model used in this work was first resented by Sun et at. [1,]. This model was develoed based on the one-arameter lastic model roosed by Chen and Sun [7]. This model was very successful in reresenting the non-linear behavior of fiberreinforced olymeric comosite materials. The requirements of orthotroic symmetry and assumtion that in the fiber direction the material is linear elastic leaded to a very simle otential function for lane stress state f 1 = σ + a66σ 1 (1) where a 66 is an orthotroy coefficient and σ σ 1 are the transverse and shear stresses, resectively. It is assumed that strain increments are so small that can be searated into elastic and lastic arts as e ij ij ij d = d + d () The elastic art follows elastic strain/stress relations for the comosite and the lastic strains are obtained from the otential function as f dij = dλ σ ij (3) Defining the effective stress as 3 σ = 3 f = + ( σ a ) 66 1 σ (4) and using the equivalence of lastic work rate [7], the effective lastic strain rate is obtained as ( ) ( 1 ) 1 & = & + & γ (5) 3 a66 where & and γ& 1 are the transverse and shear lastic strain rate, resectively. Under constant strain rate, the effective lastic strain can be related with uniaial alied stress σ and lastic strain as σ = (6) h ( θ) σ = (7) h ( θ ) where h(θ) is an off-ais arameter defined as

3 h = 3 sen + sen cos θ (8) 4 ( θ ) ( θ a66 θ ) The reference master curve effective lastic strain/effective stress is obtained for the transverse direction, since for this direction h(θ) do not deendent on the orthotroy coefficient a 66. The coefficient a 66 should be such that all curves obtained for all other directions collase into a single master curve, i.e. the reference master curve. Therefore the viscolastic model assumes that for each strain rate there is a unique master curve which can be reresented by a ower law as A( ) n = σ (9) where n is a material constant and the arameter A is function of the effective lastic strain rate. Sun et al. [1,] roosed the following relationshi ( ) m A = χ & (1) where χ e m are material constants. The off-ais material resonse for the unidirectional loading tests can be redicted using the following non-linear equation & σ & & (11) = + E where E is the aarent modulus of elasticity of the off-ais secimen which is obtained from the transformation equation = + + E E G E E 4 υ1 4 cos θ sin θ cos θ sin θ (1) where E 1, E, G 1 and ν 1 are the orthotroic elastic constants for the fiber comosite. After algebraic maniulation of Equation (11) a convenient non-linear differential equation can be obtained, Sun and Thiruukuzhi [], using equations (6) and (7) and the eressions and from equations (9) and (1) 1 & = χ ( ) & & (13) = h θ 1 m nm ( σ) ( ) Finally the aroriate non-linear differential equation is obtained 1 m 1 m 1 nm 1 m nm ( ) ( ) ( ) (14) 1 1 m & σ + E h θ σ E = χ & (15)

4 where σ = (16) E It is interesting to note that revious equation allows cree analysis. Let us consider the tyical cree loading, with a constant load alied σ = σ for t, and = σ E for t=. Then solving equation (15) for this load condition, the following non-linear cree equation, ower law of time, is obtained where t reresents the time. m m nm m nm 1 ( ) ( ) m m m m h m χ t σ = + θ σ E (17) 3. MATERIALS & SPECIMEN PREPARATION The eoy re-reg system used in this work, Teireg HS16 REM manufactured by SEAL, is a modified eoy REM reinforced with high strength carbon fiber in the form of unidirectional tae (.15 mm thick). The laminates were roduced on a 4-ton caacity SATIM hot late ress at 13 C under 1 bar ressure during 5 minutes and 3 bar ressure during more 6 min, which gives 11 min of dwell time. The average thickness of cured laminates was.5 mm and fiber content by volume, based on the fiber contents sulied by the re-reg manufacturer, was V f.6. From the original manufactured laminates, square samles with 8.5 mm side were cut using water jet, dislayed in Figure 1. Since some of the secimens would be tested in the Slit Hokinson Pressure Bar (SHPB), it was essential to obtain an accurate arallelism between the faces that would be in contact with the bars. A secific tool was designed to erform this task. The square secimens were then laed using fine sandaer (grit #6). Fig. 1. Eerimental samles with mm 3, comared with a coin of 1 cent of euro.

5 4. EXPERIMENTAL SET-UP The low strain rate comression tests were carried out in an INSTRON conventional testing machine in dislacement controlled mode with constant cross-head seed of.1, 1 and 1 mm/min. The comression tests at high strain rate were erformed with the Slit Hokinson Pressure Bar (SHPB). The articular setu used in the current study consists of striker, incident and transmission bars made of steel. The bars diameter is 1 mm. The striker bar is 1 m long, while the incident bar length is.5 m and the transmission bar 1.5 m. The secimen was sandwiched between the incident bar and the transmission bar. Lubricant grease was alied at the secimen surfaces in contact with the bars to reduce the effect of friction and to rovide better contact. Equally, a small amount of lubricant grease was alied at the end of the striker bar to avoid high frequency henomena in the signal acquired by the oscilloscoe, consequence of heterogeneous contact. The full bridge with strain gage transducers A and B, used as signal monitors, were mounted at 15 and 15 mm from the secimen, resectively. The striker bar was released at a ressure of 1 bar by a gas gun secially made for that rooses. The transient strain history is recorded from the strain gages A and B set u on the incident and transmission bars. A PICO CM1 signal conditioner adater amlifies the gages outut signal 1 times. The data is acquired using a LeCroy 945A digital oscilloscoe at a samling rate of 1 MHz. A rogram named ADAVID [6] imorts the data from the oscilloscoe data storage to the PC for osterior analysis. 5. RESULTS & DISCUSSION An analysis of strain rate deendent mechanical behavior of unidirectional laminates was erformed. In Figures, 3, 4 and 5 the effective stress-effective lastic strain rate is lotted for three strain rates,.1,.1 and.7/s, for each off-ais secimen. In this study, it was used the viscolastic model roosed by Sun et al.[1,], reviously described. This model is very simle to use and in this case the master effective stress effective lastic strain was reresented as follows. ( σ) 5. = A (18) where the arameter A is function of the effective lastic strain rate. Assuming that A is a ower law function of effective lastic strain, it was obtained the following relationshi A ( ) = & (19) It was found that the orthotroic arameter a 66 remain the constant (a 66 =.6) for the three strain rates,.1,.1 and.7/s. In Figures, 3, 4 and 5 it is ossible to verify the model ability to reroduce eerimental data. It should be noted that the indicated strain rates are nominal values. Deending on fiber orientation real strain rates resented small deviations form nominal values. For numerical evaluation of the viscolastic model, real strain rates were used instead of nominal values. Although some discreancies are clearly observed between the model and eerimental results, the evolution of the lastic strain, function of strain rate, was cature satisfactorily for all tested orientations.

6 Effective Stress (MPa) /s - Model.1/s - Model.7/s - Model.1/s - e.1/s - e.7/s - e U - 9º Effective Plastic Strain Fig.. Comarison of effective stress/effective lastic strain curves and model fitting for transverse loading. Effective Stress (MPa) /s - Model.1/s - Model.7/s - Model.1/s - e.1/s - e.7/s - e U - 75º Effective Plastic Strain Fig. 3. Comarison of effective stress/effective lastic strain curves and model fitting for the 75º off-ais secimen. Effective Stress (MPa) U - 45º.1/s - Model.1/s - Model.7/s - Model.1/s - e.1/s - e.7/s - e Effective Plastic Strain Fig. 4. Comarison of effective stress/effective lastic strain curves and model fitting for the 45º off-ais secimen.

7 Effective Stress (MPa) U - 3º.1/s - Model.1/s - Model.7/s - Model.1/s - e.1/s - e.7/s - e Effective Plastic Strain Fig. 5. Comarison of effective stress/effective lastic strain curves and model fitting for the 3º off-ais secimen. Now, it is ossible to reconstruct the stress-strain curves for the off-ais orientations by integration of the non-linear differential equation (15). This equation describes the off-ais material resonse for the uniaial comression (or tension) tests. The Figures 6, 7, 8 and 9 ehibit model redictions in comarison with eerimental results at various strain rates. For all directions, ecet 75º off ais, the model and eerimental results are in good agreement. Further, in the case of 75º off ais, the discreancy between model and eerimental results is only verified in the case of the lower strain rate. The develoment of viscolastic model was based on low strain rates.1,.1 and.7/s. The eerimental tests for high strain rates 4/s were conducted on a Slit Hokinson Pressure Bar (SHPB) located at the Otical Lab of INEGI. At this moment there are only results for transverse direction. Still, in this direction the viscolastic model roved to be valid for high strain rates, as it is deicted in Figure 9. 5 U - 3º Stress (MPa) /s.1/s.7/s Model -.1/s Model -.7/s Model - 4/s Strain Fig. 6. Comarison of eerimental and theoretical stress-strain curves for different strain rates for the 3º offais secimen.

8 5 U - 45º Stress (MPa) /s.1/s.7/s Model -.1/s Model -.7/s Model - 4/s Strain 8 Fig. 7. Comarison of eerimental and theoretical stress-strain curves for different strain rates for the 45º offais secimen. 5 U - 75º Stress (MPa) Strain.1/s.1/s.7/s Model -.1/s Model -.7/s Model - 4/s Fig. 8. Comarison of eerimental and theoretical stress-strain curves for different strain rates for the 75º offais secimen. Stress (MPa) /s.1/s.7/s 4/s Model -.1/s Model -.7/s Model - 4/s U - 9º Strain Fig. 9. Comarison of eerimental and theoretical stress-strain curves for different strain rates for the transverse loading.

9 The 3-arameter viscolastic model was roosed by Sun and Thiruukuzhi [1,] alied and verified for GFRP comosite materials. In this work the very same model was alied to a CFRP. The first results indicate that the model ossesses a good caability to reresent the strain rate deendent mechanical behaviour of a CFRP comosite material, over a wide range of strain rates i.e. from.1 to 4/s. Therefore the 3-arameter viscolastic model aears to be alicable to CFRP comosite materials with good success. 6. CONCLUSIONS Studies were carried out on 6 quasi-isotroic and unidirectional laminates under comression loading. The setu used were one conventional mechanical testing machine (INSTRON) for strain rates between.1 and.7/s and one Slit Hokinson Pressure Bar for strain rates of about 4/s. The material system used was Teireg HS16 REM, comrising high strength carbon fiber and eoy resin. Square couons with mm 3 were used for tests. The unidirectional couon secimens were subjected to loading along the 3º, 45º, 75º and 9º directions. An attemt is made to characterize the rate-deendent non-linear behaviour of a olymeric comosite material under monotonic comression loading. A 3-arameter constitutive viscolastic model roosed by Sun and Thiruukuzhi [1,] was used to describe the mechanical behaviour. This model was develoed based on data for strain rate between.1 and.7/s. In transverse direction the viscolastic model was able to redict the high strain rate eeriments conducted on a Slit Hokinson Pressure Bar. Consequently, this 3- arameter viscolastic model was able to describe the strain rate deendent mechanical behaviour of Teireg HS16 REM comosite material, under comression, with good accuracy. ACKNOWLEDGEMENTS This work was funded by Portuguese Science and Technology Foundation (FCT) witch is artially suorted by the Euroean Community fund FEDER, under Grants POCTI/EME/448/1 and POCTI/EME/3591/. References 1. Tsai J. and Sun C.T. Constitutive model for high strain rate resonse of olymeric comosites, Comosites Science and Technology, 6 (), Thiruukuzhi S.V. and Sun C.T., Models for the strain-rate-deendent behavior of olymer comosites, Comosites Science and Technology,. 61 (1), Gates T.S., Rate Deendent Constitutive Models for Fiber Reinforced Polymer Comosites, NASA TM 1665, NASA Langley Research Center, May (199). 4. Gates T.S. and Sun C.T., An Elastic/Viscolastic Constitutive Model for Fiber Reinforced Thermolastic Comosites, AIAA Journal 9/3 (1991), Goldberg R.K. and Stouffer D.C., High Strain Rate Deformation Modeling of a Polymer Matri Comosite: Part I- Matri Constitutive Equations, NASA TM 6969, NASA Lewis Research Center, August (1998). 6. Goldberg R.K. and Stouffer D.C., Strain rate deendent analysis of a olymer matri comosite utilizing a micromechanics aroach, J. Comosite Materials 36/7 (), Sun C.T. and Chen J.L., A simle flow rule for characterizing nonlinear behavior of fiber comosites, J. of Comosite Materials, 3 (1989), Gary G. and Kleaczko, A Comuter Program for the Analysis of SHPB Test, Paris / METZ:1998