《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_c-g-7

Availableonlineatwww.sciencedirect.com COMPOSITE STRUCTURES ELSEVIER Composite Structures 63(2004)455-467 Mechanical behaviour of glass and carbon fibre reinforced composites at varying strain rates R O. Ochola . K. Marcus G.N. Nurick. t franz Centre for Materials Engineering(CME), University of Cape Town(UCT), Prirate Bag, Rondebosch 7701, South b Department of Mechanical Engineering, University of Cape Town(UCT), Prirate Bag, Rondebosch 7701, South Africa Cardiovascular Research Unit, University of Cape Town(UCT), Prirate Bag, Rondebosch 7701, South Africa Abstract The choice of composite materials as a substitute for metallic materials in technological applications is becoming more pro- nounced especially due to the great weight savings these materials offer. In many of these practical situations, the structures are prone to high impact loads. Material and structural response vary significantly under impact loading conditions as compared to uasi-static loading. The strain rate sensitivity of both carbon fibre reinforced polymer(CFRP)and glass fibre reinforced polymer GFRP)are studied by testing a single laminate configuration, viz. cross-ply [0/90] polymer matrix composites(PMc)at strain ites of 10' and 450 s. The compressive material properties are determined by testing both laminate systems, viz. CFRP and GFRP at low to high strain rates. The laminates were fabricated from 48 layers of cross-ply carbon fibre and glass fibre epoxy Dynamic test results were compared with static compression test carried out on specimens with the same dimensions. Preliminary compressive stress-strain vs strain rates data obtained show that the dynamic material strength for GFRP increases with increasing strain rates. The strain to failure for both CfrP and gfrp is seen to decrease with increasing strain rate C 2003 Elsevier Ltd. All rights reserved. Keywords: Strain rate sensitivity: Split Hopkinson bar technique; CFRP: GFRP; Dynamic test 1. Introduction could be used to explain the variation in material strength, stress and strain with varying strain rates Polymer matrix composites(PMC) have material While metals have been studied extensively over a wide properties which are attractive for use in various engi- range of strain rates, limited information is available tiff- with regard to the effect of strain rates on fibrous com ness and strength, fatigue properties, and corrosion posites. The majority of information concerning strain resistance make these materials especially appealing to rate sensitivity of fibrous composites at rates above 100 the aerospace, civil engineering, marine and automobile s-I was obtained from tests involving projectile impact industries. Many of the applications require service as summarised by Sierakowski and Nevill [1]. However under dynamic loading conditions. In the aerospace such tests may not easily be used to determine the ma dustry primary aircraft structures such as wings and terial constitutive relations since the state of stress and impact from bird strikes or foreign objects. Automotive, easily monitored during impale orm and the stress is not turbine blades are prone to experience high velocity strain in the specimen is not uni marine and civil structures in service are also prone to At high rates of stra high velocity impact from foreign bodies. Therefore the consequently wave propagation effects become a factor, understanding of the material response under dynamic as a result the experimental technique employed must in loading becomes imperative. This is achieved through some form involve the propagation of stress waves. One experimental and theoretical analysis. The latter re- of the most extensively used experimental configurations quires the development of constitutive equations, which for high strain rate material characterisation is the Hopkinson pressure bar or Kolsky bar [2]. The concept of the Hopkinson bar involves the determination of the E-mail dres ochroboo1@ mail ct ac a (R.O. Ochola dynamic stresses, strains or displacements occurring at he end of a bar through monitoring the elastic pulses in 0263-8223/S- see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)001946

Mechanical behaviour of glass and carbon fibre reinforced composites at varying strain rates R.O. Ochola a,*, K. Marcus a , G.N. Nurick b , T. Franz c a Centre for Materials Engineering (CME), University of Cape Town (UCT), Private Bag, Rondebosch 7701, South Africa b Department of Mechanical Engineering, University of Cape Town (UCT), Private Bag, Rondebosch 7701, South Africa c Cardiovascular Research Unit, University of Cape Town (UCT), Private Bag, Rondebosch 7701, South Africa Abstract The choice of composite materials as a substitute for metallic materials in technological applications is becoming more pro￾nounced especially due to the great weight savings these materials offer. In many of these practical situations, the structures are prone to high impact loads. Material and structural response vary significantly under impact loading conditions as compared to quasi-static loading. The strain rate sensitivity of both carbon fibre reinforced polymer (CFRP) and glass fibre reinforced polymer (GFRP) are studied by testing a single laminate configuration, viz. cross-ply [0
/90
] polymer matrix composites (PMC) at strain rates of 10
3 and 450 s
1. The compressive material properties are determined by testing both laminate systems, viz. CFRP and GFRP at low to high strain rates. The laminates were fabricated from 48 layers of cross-ply carbon fibre and glass fibre epoxy. Dynamic test results were compared with static compression test carried out on specimens with the same dimensions. Preliminary compressive stress–strain vs. strain rates data obtained show that the dynamic material strength for GFRP increases with increasing strain rates. The strain to failure for both CFRP and GFRP is seen to decrease with increasing strain rate. 2003 Elsevier Ltd. All rights reserved. Keywords: Strain rate sensitivity; Split Hopkinson bar technique; CFRP; GFRP; Dynamic test 1. Introduction Polymer matrix composites (PMC) have material properties which are attractive for use in various engi￾neering applications. Factors such as high specific stiff- ness and strength, fatigue properties, and corrosion resistance make these materials especially appealing to the aerospace, civil engineering, marine and automobile industries. Many of the applications require service under dynamic loading conditions. In the aerospace industry primary aircraft structures such as wings and turbine blades are prone to experience high velocity impact from bird strikes or foreign objects. Automotive, marine and civil structures in service are also prone to high velocity impact from foreign bodies. Therefore the understanding of the material response under dynamic loading becomes imperative. This is achieved through experimental and theoretical analysis. The latter re￾quires the development of constitutive equations, which could be used to explain the variation in material strength, stress and strain with varying strain rates. While metals have been studied extensively over a wide range of strain rates, limited information is available with regard to the effect of strain rates on fibrous com￾posites. The majority of information concerning strain rate sensitivity of fibrous composites at rates above 100 s
1 was obtained from tests involving projectile impact as summarised by Sierakowski and Nevill [1]. However, such tests may not easily be used to determine the ma￾terial constitutive relations since the state of stress and strain in the specimen is not uniform and the stress is not easily monitored during impact. At high rates of strain ( P10
2 s
1), inertia and consequently wave propagation effects become a factor, as a result the experimental technique employed must in some form involve the propagation of stress waves. One of the most extensively used experimental configurations for high strain rate material characterisation is the Hopkinson pressure bar or Kolsky bar [2]. The concept of the Hopkinson bar involves the determination of the dynamic stresses, strains or displacements occurring at the end of a bar through monitoring the elastic pulses in * Corresponding author. Fax: +27-21-689-7571. E-mail address: ochrob001@mail.uct.ac.za (R.O. Ochola). 0263-8223/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00194-6 Composite Structures 63 (2004) 455–467 www.elsevier.com/locate/compstruct

RO. Ochola et al. Composite Structures 63(2004)455-467 Nomenclature strain rate yield stress of bars material density of Hopk E Young,s modulus of bars length of specimen time od dynamic ultimate stress kink angle D diameter of specimen number of standard deviations quasi-static ultimate stress L/D length to diameter ratio X expected stress values stress in incident bar average stress values stress in transmitter bar original specimen length reflected stress in incident bar final specimen length Umax striker velocity final velocity longitudinal wave velocit the bar some distance away. The basic principle requires depend on various composite constituent properties, that the bar remains elastic, and as a result the distur- geometric arrangement, volume fraction of reinforce- bances at the end of the bar are propagated undistorted ment as well as the size of the reinforcing fibres Siera- along the bar at the elastic wave velocity. The Hopkin- kowski and Nevill [ l] found a simple energy criterion for son bar was first developed in 1914 by Hopkinson [2] composite systems with a brittle matrix for predicting who used the induced wave propagation in a long elastic delamination Sierakowski and Nevill,s [1] work showed metal bar to measure the pressure produced during a that the high strain rate or impact failure characteristics dynamic event. Through the use of momentum traps of are often distinctly different from failure observed at low differing length, Hopkinson was able to study the evo- strain rate testing. Griffiths and Martin [6] investigated lution and the shape of stress pulses as a function of time the dynamic behaviour of unidirectional carbon fibre as they propagated along the metal bar. Davies [3] and composites at high strain rates to determine how the Kolsky [4]. working independently later, used two material behaviour is dependent on fibre volume frac Hopkinson pressure bars in series with the sample tion and fibre orientation. Results from these studies sandwiched between the bars, to measure the dynamic indicate that the dynamic stress-strain characteristics of stress-strain response of materials. Their work formed carbon-fibre composites in compression differ consid- the basis of the split Hopkinson pressure bar currently erably from those under static loading, however no high strain rate research. Johns and strain rate effects are noted for changes in strain rate Cook [5] employed a split Hopkinson pressure bar for between 320 and 550 s. It was also noted that the re- determining the material properties of metals duction in the elastic modulus at high strains rates ap- The experience of the use of split Hopkinson pressure pears to be due to the specimen geometry and is bar for the investigation of metals led to the adaptation therefore not an intrinsic property of the composite of this technique for the characterization of laminated The dynamic compressive behaviour of unidirectional polymer composites at strain rates ranging from 10 to and transversely isotropic glass-epoxy composite are P s. However, only in the recent past has significant determined by Kumar et al. [7, using the Kolsky pressure efforts been made to examine the high strain rate be- bar technique for six fibre orientations, viz. 0, 10, 300 haviour using the split Hopkinson bar, of more brittle 45, 60 and 90 at a strain rate, E, of 265(+50)s materials such as composites and ceramics, to measure Studies were carried out on cylindrical specimens of high strain rate response of materials under varying varying lengths 12-35 mm, and diameters 16-17 mm, loading conditions( tension, compression, shear and respectively. Quasi-static stress-strain curves at strain torsion [1, 7-9D rates of 2x- are compared to corresponding dy The focus of this paper is on the compression re- namic curves. The results indicate that for all orienta- ponse of fibrous composites. Early research on high tions of glass epoxy there is a change in the failure modes, strain rate response of laminate composites dates back as well as an increase in strength at higher strain rates almost three decades. Sierakowski and Nevill [l]in 1971 Specimens of 00 orientation fracture along the fibres by investigated the compressive strength and failure of steel tensile splitting, due to the compressive loads causing reinforced epoxy composites. The investigation involved transverse tensile strains because of Poisson's effect mechanical testing at strain rates ranging from 10- to Therefore, as the transverse tensile strain exceeds the 104 s-l. For the composites tested, strain rates effects transverse ultimate strain, failure occurs by tensile spli

the bar some distance away. The basic principle requires that the bar remains elastic, and as a result the distur￾bances at the end of the bar are propagated undistorted along the bar at the elastic wave velocity. The Hopkin￾son bar was first developed in 1914 by Hopkinson [2] who used the induced wave propagation in a long elastic metal bar to measure the pressure produced during a dynamic event. Through the use of momentum traps of differing length, Hopkinson was able to study the evo￾lution and the shape of stress pulses as a function of time as they propagated along the metal bar. Davies [3] and Kolsky [4], working independently later, used two Hopkinson pressure bars in series with the sample sandwiched between the bars, to measure the dynamic stress–strain response of materials. Their work formed the basis of the split Hopkinson pressure bar currently employed in high strain rate research. Johnson and Cook [5] employed a split Hopkinson pressure bar for determining the material properties of metals. The experience of the use of split Hopkinson pressure bar for the investigation of metals led to the adaptation of this technique for the characterization of laminated polymer composites at strain rates ranging from 10 to 104 s
1. However, only in the recent past has significant efforts been made to examine the high strain rate be￾haviour using the split Hopkinson bar, of more brittle materials such as composites and ceramics, to measure high strain rate response of materials under varying loading conditions (tension, compression, shear and torsion [1,7–9]). The focus of this paper is on the compression re￾sponse of fibrous composites. Early research on high strain rate response of laminate composites dates back almost three decades. Sierakowski and Nevill [1] in 1971 investigated the compressive strength and failure of steel reinforced epoxy composites. The investigation involved mechanical testing at strain rates ranging from 10
5 to 104 s
1. For the composites tested, strain rates effects depend on various composite constituent properties, geometric arrangement, volume fraction of reinforce￾ment as well as the size of the reinforcing fibres. Siera￾kowski and Nevill [1] found a simple energy criterion for composite systems with a brittle matrix for predicting delamination. Sierakowski and Nevill
s [1] work showed that the high strain rate or impact failure characteristics are often distinctly different from failure observed at low strain rate testing. Griffiths and Martin [6] investigated the dynamic behaviour of unidirectional carbon fibre composites at high strain rates to determine how the material behaviour is dependent on fibre volume frac￾tion and fibre orientation. Results from these studies indicate that the dynamic stress–strain characteristics of carbon–fibre composites in compression differ consid￾erably from those under static loading, however no strain rate effects are noted for changes in strain rate between 320 and 550 s
1. It was also noted that the re￾duction in the elastic modulus at high strains rates ap￾pears to be due to the specimen geometry and is therefore not an intrinsic property of the composite. The dynamic compressive behaviour of unidirectional and transversely isotropic glass–epoxy composite are determined by Kumar et al. [7], using the Kolsky pressure bar technique for six fibre orientations, viz. 0
, 10
, 30
, 45
, 60
and 90
at a strain rate, e_, of 265(±50) s
1. Studies were carried out on cylindrical specimens of varying lengths 12–35 mm, and diameters 16–17 mm, respectively. Quasi-static stress–strain curves at strain rates of 2 · 10
4 s
1 are compared to corresponding dy￾namic curves. The results indicate that for all orienta￾tions of glass epoxy there is a change in the failure modes, as well as an increase in strength at higher strain rates. Specimens of 0
orientation fracture along the fibres by tensile splitting, due to the compressive loads causing transverse tensile strains because of Poisson
s effect. Therefore, as the transverse tensile strain exceeds the transverse ultimate strain, failure occurs by tensile split￾Nomenclature e_ strain rate q material density of Hopkinson bars L length of specimen rd dynamic ultimate stress D diameter of specimen rs quasi-static ultimate stress L=D length to diameter ratio ri stress in incident bar rt stress in transmitter bar rr reflected stress in incident bar v velocity of transmitted pulse vmax striker velocity C0 longitudinal wave velocity ry yield stress of bars E Young
s modulus of bars t time b kink angle s number of standard deviations SD standard deviation Xexpected expected stress values X average stress values l0 original specimen length l final specimen length v1 initial velocity v2 final velocity 456 R.O. Ochola et al. / Composite Structures 63 (2004) 455–467

RO. Ochola et al. Composite Structures 63 (2004)455-467 ting. Specimens of 10, 30 and 45 orientation fracture distinct points will shift. The ultimate stress increases along the fibre predominantly by interlaminar shear, al- and the point at which complete separation takes place though cracks caused by a degree of tensile splitting are also seen on the surface of some of the specimens. Kumar delay) in the mode of failure at increasing strain rates et al. [7] deduced that the dynamic stress-strain curves Conclusion from this work also state that the test are linear up to fracture for angles of0 and 100, and non- specimens should maintain the same length to diameter linear for orientations greater than 100 ratio, L/D=l, and that the ultimate compressive stress Most of the research work in the area of dynamic is not influenced by the strain rate for values of &< 100 compressive testing of PMC has, however, been under- s-I but varies linearly with the strain rate for values of taken in the last decade. Work in 1991 by El-Habak [8] 100 <E< 1000 s-. Inherently the strain rate sensitivity investigated the behaviour of woven glass fibre rein- was found to be dependent on the matrix type and the forced composites subjected fibre volume fraction loading. The compression split Hopkinson bar was usec Tay et al. [12] studied specimens of cross-woven gl to produce failure at strain rates ranging from 10 to 10 fibre reinforced epoxy and pure epoxy resin was sub- s-l Results illustrated a slight increase in the compres- jected to dynamic compressive loading. Quasi-static sive strength for all composite variables such as fibre (E 10-3s-)and low strain rate(E 10 s-)tests were orientation and fibre volume fraction. The research conducted on a hydraulic tester, while a split Hopkinson concentrated on the comparison of selected matrix sys- pressure bar was used for tests at dynamic strain rates tems viz. vinyl ester, polyester and epoxy. The highest (E 10s"). Tay et al. [12] therefore proposed a simple strength is obtained from the composite based on vinyl empirical equation based on experimental results that ester matrix. Harding [9] investigated the effect of strain showed the behaviour of glass fibre reinforced polymer rate and specimen geometry of woven glass reinforced (GFRP) and pure epoxy to be strain rate sensitive. The epoxy laminate on two different specimen designs: (1)a non-linear stress-strain behaviour is inherently a matrix- olid cylinder, as commonly used with the compression dominated property, therefore the same form of equa- version of the Hopkinson bar and (2)a thin strip, wa- tion developed is applicable to both pure epoxy and isted in the thickness direction, as generally recom- GFRP. Unidirectional glass fibre reinforced polymer mended for composite laminate compression testing. composites are investigated by Takeda and Wan [13] Mean stress-strain curves for both specimen geometries using the improved SHPB apparatus that eliminates indicated a significant increase in the initial modulus, multiple loading of the compressive specimen, which is strength and the strain at failure with increasing strain prevalent in the conventional SHPB. Scanning electron rate. Generally under compression loading the shear microscopic analysis of the tested specimens revealed resistance of the epoxy resin matrix is likely to con- that the compressive failure of the unidirectional GFRP tribute considerably to both the elastic properties of t is caused by micro-buckling of the fibres under both laminate and to the resistance to shear band formation. impact and static loading conditions. The compressive The mechanical propert of epoxy resin are known strength is related to the non-linear in-plane shear be strongly rate dependent [10], therefore the effect of modulus of unidirectional composites, and the theoret strain rate on the initial Young,s modulus and the ulti- ical prediction shows good agreement with the experi- mate compressive strength is detected mental results. These results show an increase in El-Habak [11] studied the effects of various composite compressive strength with increasing strain rates [13] properties such as fibre volume fraction and specimen The compressive behaviour of unidirectional carbon/ size at high strain rates on glass fibre in a polyester and epoxy composites was investigated at varying strain epoxy resin, respectively. Results illustrated in the form rates by Hsiao and Daniel [14]. The transverse com- stress-strain and stress vs. strain rate curves show pressive strength increases at dynamic strain rates, to three distinct regions. The first being the maximum approximately twice the quasi-static value [14. The ul values of the strain rate achieved while the glass epoxy timate strain shows no strain rate effect, which implies a pecimen is being deformed elastically. The corre- strain dominated failure criterion for analysis under ponding value on the stress curve may be expressed as dynamic loadings. Longitudinal and cross-ply com- the elastic limit. The next region represents plastic de- pressive properties were obtained for varying strain formation. The final point represents the ultimate stress rates. The results show increases in strength and ultimate and the corresponding value of strain rate. This point on strain but only a moderate increase in initial elastic the stress-strain curve represents the maximum resis- modulus. Studies have also been carried out on ofl-axis tance of the specimen to fracture. An inflection point thermoplastic composite laminate specimens, which were exists beyond the final point on the stress-strain curve, tested at a wide range of strain rates. It was found that the after which the stress decreases more rapidly. This point composite behaviour is elastic up to failure in the fibre may be taken as the start of complete separation be- direction. Significant non-linear and strain rate depen tween the fibres and matrix. At higher strain rates these dent behaviour is exhibited by off-axis and angle-ply

ting. Specimens of 10
, 30
and 45
orientation fracture along the fibre predominantly by interlaminar shear, al￾though cracks caused by a degree of tensile splitting are also seen on the surface of some of the specimens. Kumar et al. [7] deduced that the dynamic stress–strain curves are linear up to fracture for angles of 0
and 10
, and non￾linear for orientations greater than 10
. Most of the research work in the area of dynamic compressive testing of PMC has, however, been under￾taken in the last decade. Work in 1991 by El-Habak [8] investigated the behaviour of woven glass fibre rein￾forced composites subjected to compressive impact loading. The compression split Hopkinson bar was used to produce failure at strain rates ranging from 10 to 103 s
1. Results illustrated a slight increase in the compres￾sive strength for all composite variables such as fibre orientation and fibre volume fraction. The research concentrated on the comparison of selected matrix sys￾tems viz. vinyl ester, polyester and epoxy. The highest strength is obtained from the composite based on vinyl ester matrix. Harding [9] investigated the effect of strain rate and specimen geometry of woven glass reinforced epoxy laminate on two different specimen designs: (1) a solid cylinder, as commonly used with the compression version of the Hopkinson bar and (2) a thin strip, wa￾isted in the thickness direction, as generally recom￾mended for composite laminate compression testing. Mean stress–strain curves for both specimen geometries indicated a significant increase in the initial modulus, strength and the strain at failure with increasing strain rate. Generally under compression loading the shear resistance of the epoxy resin matrix is likely to con￾tribute considerably to both the elastic properties of the laminate and to the resistance to shear band formation. The mechanical properties of epoxy resin are known to be strongly rate dependent [10], therefore the effect of strain rate on the initial Young
s modulus and the ulti￾mate compressive strength is detected. El-Habak [11] studied the effects of various composite properties such as fibre volume fraction and specimen size at high strain rates on glass fibre in a polyester and epoxy resin, respectively. Results illustrated in the form of stress–strain and stress vs. strain rate curves show three distinct regions. The first being the maximum values of the strain rate achieved while the glass epoxy specimen is being deformed elastically. The corre￾sponding value on the stress curve may be expressed as the elastic limit. The next region represents plastic de￾formation. The final point represents the ultimate stress and the corresponding value of strain rate. This point on the stress–strain curve represents the maximum resis￾tance of the specimen to fracture. An inflection point exists beyond the final point on the stress–strain curve, after which the stress decreases more rapidly. This point may be taken as the start of complete separation be￾tween the fibres and matrix. At higher strain rates these distinct points will shift. The ultimate stress increases and the point at which complete separation takes place are delayed. El-Habak [8] therefore shows a change (or delay) in the mode of failure at increasing strain rates. Conclusion from this work also state that the test specimens should maintain the same length to diameter ratio, L=D ¼ 1, and that the ultimate compressive stress is not influenced by the strain rate for values of e_ < 100 s
1 but varies linearly with the strain rate for values of 100 < e_ < 1000 s
1. Inherently the strain rate sensitivity was found to be dependent on the matrix type and the fibre volume fraction. Tay et al. [12] studied specimens of cross-woven glass fibre reinforced epoxy and pure epoxy resin was sub￾jected to dynamic compressive loading. Quasi-static (e_ 10
3 s
1) and low strain rate (e_ 10 s
1) tests were conducted on a hydraulic tester, while a split Hopkinson pressure bar was used for tests at dynamic strain rates (e_ 103 s
1). Tay et al. [12] therefore proposed a simple empirical equation based on experimental results that showed the behaviour of glass fibre reinforced polymer (GFRP) and pure epoxy to be strain rate sensitive. The non-linear stress–strain behaviour is inherently a matrix￾dominated property, therefore the same form of equa￾tion developed is applicable to both pure epoxy and GFRP. Unidirectional glass fibre reinforced polymer composites are investigated by Takeda and Wan [13] using the improved SHPB apparatus that eliminates multiple loading of the compressive specimen, which is prevalent in the conventional SHPB. Scanning electron microscopic analysis of the tested specimens revealed that the compressive failure of the unidirectional GFRP is caused by micro-buckling of the fibres under both impact and static loading conditions. The compressive strength is related to the non-linear in-plane shear modulus of unidirectional composites, and the theoret￾ical prediction shows good agreement with the experi￾mental results. These results show an increase in compressive strength with increasing strain rates [13]. The compressive behaviour of unidirectional carbon/ epoxy composites was investigated at varying strain rates by Hsiao and Daniel [14]. The transverse com￾pressive strength increases at dynamic strain rates, to approximately twice the quasi-static value [14]. The ul￾timate strain shows no strain rate effect, which implies a strain dominated failure criterion for analysis under dynamic loadings. Longitudinal and cross-ply com￾pressive properties were obtained for varying strain rates. The results show increases in strength and ultimate strain but only a moderate increase in initial elastic modulus. Studies have also been carried out on off-axis thermoplastic composite laminate specimens, which were tested at a wide range of strain rates. It was found that the composite behaviour is elastic up to failure in the fibre direction. Significant non-linear and strain rate depen￾dent behaviour is exhibited by off-axis and angle-ply R.O. Ochola et al. / Composite Structures 63 (2004) 455–467 457

R.O. Ochola et al. Composite Structures 63(2004)455-467 laminates. Comparison of the dynamic stress-strain curves with the quasi-static stress-strain curve shows the &s au f fects of strain rate, as noted by Weeks and Sun [15] There is no standardized test specimen geometry for Differentiating the strain in Eq.(2)with respect to the tests on composites with the SHPB. An attempt by time, t, gives the velocity investigate the effects of varying the length to diameter t au Woldesenbet and Vinson [16 was therefore made to at spect to the material properties at varying strain rates of where f" denotes the differentiation of f with respect to between 4x102 and 1.3x 103 s-1 The results show no x. The velocity in terms of the strain, a, and wave ve- statistically significant effect of either L/D or geometry locity, Co, is given by for carbon/epoxy laminates tested at varying strain rates. Both specimen shapes tested result in similar high (4) strain rate properties, therefore comparison can be made between varying test specimen shapes Gary and Zhao [17] employed the use of low im- pedance material such as nylon, for the incident and transmitter bars of the split Hopkinson bar, to test the substituting a=Ee and E= Cop, Eq. (5)can be re- strain rate behaviour of glass epoxy composite plates written as The failure strength of the glass epoxy plate tested by Gary and Zhao [17] is reported to be strain rate sensi U=-COE pCo tive. Fibre orientation effects on high strain rate prop. erties were considered recently for a carbon epoxy system. Vinson and Woldensenbet's [18] results show that the ultimate stress increases with increasing strain rate. Attempts to characterize the high strain rate be- where p is the density of the bar. haviour of fibl off-axis laminates and Due to convention in wave theory the negative sign in he split Hopkinson bar were carried out by Ninan et al Eq.(7) indicates the direction of the tensile [19]. Increases in the stress values of the off-axis glass therefore in this case the tensile pulse moves poxy composites are noted when the strain rate of the positive x direction (Fig. I). The compressive compressive loads is increased from static to dynamic taken as positive can be deduced from for the varying fibre orientations. Hosur et al. [20]in- o=pC vestigated the response of carbon/epoxy laminated composites under high strain rate compression loading The stress at the incident bar/specimen interface The results from this study indicate that the dynamic lown as A in Fig. 1, is given by strength and youngs modulus exhibit considerable in creases as compared to the static value (a-Or)ABar The objective of this paper is to investigate the com- pressive behaviour on cylindrical specimens of cross-ply and the stress at the specimen/transmitter bar interface, GFRP and carbon fibre reinforced polymer( CFRP) at labelled B in Fig. 1, is given by quasi-static and dynamic strain rates using a hydraulic compression rig and the split Hopkinson bar. A const utive relation is developed from the experimental data a B lo 2. Hopkinson bar theory a wave propagating in an elastic bar in the positive x direction is considered. The general solution of the lementary wave equation theory [23] becomes ncident bar Transmitter bar u=f(x-Cot) where u is the displacement, t the time and x the di placement of the particle. The one-dimensional strain is determined by differentiating the displacement equation Fig. 1 Schematic diagram of a compressive split Hopkinson pressure (1) with respect to x to give

laminates. Comparison of the dynamic stress–strain curves with the quasi-static stress–strain curve shows the effects of strain rate, as noted by Weeks and Sun [15]. There is no standardized test specimen geometry for tests on composites with the SHPB. An attempt by Woldesenbet and Vinson [16] was therefore made to investigate the effects of varying the length to diameter (L=D) ratio and/or geometry of the specimen with re￾spect to the material properties at varying strain rates of between 4 · 102 and 1.3 · 103 s
1. The results show no statistically significant effect of either L=D or geometry for carbon/epoxy laminates tested at varying strain rates. Both specimen shapes tested result in similar high strain rate properties, therefore comparison can be made between varying test specimen shapes. Gary and Zhao [17] employed the use of low im￾pedance material such as nylon, for the incident and transmitter bars of the split Hopkinson bar, to test the strain rate behaviour of glass epoxy composite plates. The failure strength of the glass epoxy plate tested by Gary and Zhao [17] is reported to be strain rate sensi￾tive. Fibre orientation effects on high strain rate prop￾erties were considered recently for a carbon epoxy system. Vinson and Woldensenbet
s [18] results show that the ultimate stress increases with increasing strain rate. Attempts to characterize the high strain rate be￾haviour of fibre composites using off-axis laminates and the split Hopkinson bar were carried out by Ninan et al. [19]. Increases in the stress values of the off-axis glass epoxy composites are noted when the strain rate of the compressive loads is increased from static to dynamic for the varying fibre orientations. Hosur et al. [20] in￾vestigated the response of carbon/epoxy laminated composites under high strain rate compression loading. The results from this study indicate that the dynamic strength and Young
s modulus exhibit considerable in￾creases as compared to the static values. The objective of this paper is to investigate the com￾pressive behaviour on cylindrical specimens of cross-ply GFRP and carbon fibre reinforced polymer (CFRP) at quasi-static and dynamic strain rates using a hydraulic compression rig and the split Hopkinson bar. A constit￾utive relation is developed from the experimental data. 2. Hopkinson bar theory A wave propagating in an elastic bar in the positive x direction is considered. The general solution of the elementary wave equation theory [23] becomes u ¼ f ðx
C0tÞ ð1Þ where u is the displacement, t the time and x the dis￾placement of the particle. The one-dimensional strain is determined by differentiating the displacement equation (1) with respect to x to give e ¼ ou ox ¼ f 0 ð2Þ Differentiating the strain in Eq. (2) with respect to the time, t, gives the velocity v ¼ ou ot ¼
C0f 0 ð3Þ where f 0 denotes the differentiation of f with respect to x. The velocity in terms of the strain, e, and wave ve￾locity, C0, is given by ou ot ¼
C0 ou ox ð4Þ or v ¼
C0e ð5Þ substituting r ¼ Ee and E ¼ C2 0q, Eq. (5) can be re￾written as v ¼
C0 r E ¼
r qC0 ð6Þ giving r ¼
qC0v ð7Þ where q is the density of the bar. Due to convention in wave theory the negative sign in Eq. (7) indicates the direction of the tensile pulse, therefore in this case the tensile pulse moves in the positive x direction (Fig. 1). The compressive stresses taken as positive can be deduced from r ¼ qC0v ð8Þ The stress at the incident bar/specimen interface, shown as A in Fig. 1, is given by rs1 ¼ ðri
rrÞABar AsðtÞ ð9Þ and the stress at the specimen/transmitter bar interface, labelled B in Fig. 1, is given by Fig. 1. Schematic diagram of a compressive split Hopkinson pressure bar. 458 R.O. Ochola et al. / Composite Structures 63 (2004) 455–467

R.O. Ochola et al. Composite Structures 63(2004)455-467 Ct 032=A(0) (10) mens used in both the SHPB and the hydraulic machine are shown in Fig. 2. The displacement in the two bars results in true strain of 3.2. Quasi-static compressive tests the specimen A hydraulic testing machine was used to determine In the compressive strengths of carbon and glass fibre specimens at quasi-static strain rates. The specimens By differentiating Eq.(11)with respect to time, t, the were tested between two flat metal cylinders one at- strain rate in the specimen is given as tached to the grips and the second at the load cell. The U1-U2 assembly was placed so that the specimens were cen- o(1) (12) trally located between the metal stages, and compressed at a rate of 0.5 mm/min, equivalent to a strain rate of 2x10-'s. The quasi-static results were recorded in the form of load displacement curves with maximum loads 3. Experimental set-up and procedure of 30 kN for displacements of 0.4-0.7 mm. This data converted into stress-strain curves to determine the 3.1. Materials and specimens strain and ultimate compressive strengths. The quasi static experimental set-up was designed so as to enable a In this study 48-ply layered cross-ply laminates were direct comparison of the compressive material strengtH manufactured for the purpose of investigating strain parameters from the quasi-static tests with those of the rate effects. The gfRP laminates were fabricated from split Hopkinson bar. The focus is on keeping constant unidirectional (UD)standard E glass in an Ampreg 20 specimen geometry for both sets of tests viz. the quasi epoxy matrix. The CFRP was fabricated using UT static and dynamic strain rates C300/500 carbon tows in an epoxy matrix. Laminates in the form of plates of thickness 10.8 mm were cured at 60 3.3. High strain rate compressive tests for 2 h. The plates were machined into specimens with diameter (D)8.0 mm and length 8.0 mm. The For the high strain rate test a split Hopkinson bar set lengths (L)are reduced to 4.0 mm from subsequent up was employed. A schematic of the set-up is shown in polishing of the specimen. Extreme care was exercised Fig 3. The set-up comprises a gas gun/test chamber(D), when shaping the specimens to the desired geometry to striker bar (2), incident bar (4), transmitter bar (6). ensure dimensional accuracy and to prevent delamina- strain gauges(3i, 1), on incident bar and transmitter bar tions in the composite specimens due to machining. The and momentum trap or stopper(7). In the SHPB tech dimensions, geometry and nique a small st elastic bars (4 and 6) of similar cross-sectional area and Youngs modulus. The test chamber pressure of the Hopkinson bar varied from 400 to 600 kPa, results in strain rates of between 200 and 900 s-. An elastic stress pulse is imparted into the incident bar (4) by impacting it with a striker bar(2). The impact of the striker bar generates a longitudinal compressive incident stress L pulse, i, equal to twice the length of the striker bar, that travels down the incident bar and is recorded by the pecten Geometry Direction strain gauge(3i) at the incident bar. The pulse then reaches the incident bar/specimen interface where part Fig. 2. Geometry and loading direction for specimens. of the pulse is reflected in the form of a tensile stress I Gas gun 2 Striker bar 3 Strain gauges 4 Incident bar 5 Specimen 6 Transmitter bar Fig. 3. Schematic diagram of a compressive split Hopkinson pressure bar[23]

rs2 ¼ rtABar AsðtÞ ð10Þ The displacement in the two bars results in true strain of the specimen etrue ¼ ln l l0
ð11Þ By differentiating Eq. (11) with respect to time, t, the strain rate in the specimen is given as e_ ¼ v1
v2 l0ðtÞ ð12Þ 3. Experimental set-up and procedure 3.1. Materials and specimens In this study 48-ply layered cross-ply laminates were manufactured for the purpose of investigating strain rate effects. The GFRP laminates were fabricated from unidirectional (UD) standard E glass in an Ampreg 20 epoxy matrix. The CFRP was fabricated using UT￾C300/500 carbon tows in an epoxy matrix. Laminates in the form of plates of thickness 10.8 mm were cured at 60
C for 2 h. The plates were machined into specimens with diameter (D) 8.0 mm and length 8.0 mm. The lengths (L) are reduced to 4.0 mm from subsequent polishing of the specimen. Extreme care was exercised when shaping the specimens to the desired geometry to ensure dimensional accuracy and to prevent delamina￾tions in the composite specimens due to machining. The dimensions, geometry and loading directions of the specimens used in both the SHPB and the hydraulic machine are shown in Fig. 2. 3.2. Quasi-static compressive tests A hydraulic testing machine was used to determine the compressive strengths of carbon and glass fibre specimens at quasi-static strain rates. The specimens were tested between two flat metal cylinders one at￾tached to the grips and the second at the load cell. The assembly was placed so that the specimens were cen￾trally located between the metal stages, and compressed at a rate of 0.5 mm/min, equivalent to a strain rate of 2 · 10
3 s
1. The quasi-static results were recorded in the form of load displacement curves with maximum loads of 30 kN for displacements of 0.4–0.7 mm. This data is converted into stress–strain curves to determine the strain and ultimate compressive strengths. The quasi￾static experimental set-up was designed so as to enable a direct comparison of the compressive material strength parameters from the quasi-static tests with those of the split Hopkinson bar. The focus is on keeping constant specimen geometry for both sets of tests viz. the quasi￾static and dynamic strain rates. 3.3. High strain rate compressive tests For the high strain rate test a split Hopkinson bar set￾up was employed. A schematic of the set-up is shown in Fig. 3. The set-up comprises a gas gun/test chamber (1), striker bar (2), incident bar (4), transmitter bar (6), strain gauges (3i; t), on incident bar and transmitter bar and momentum trap or stopper (7). In the SHPB tech￾nique a small specimen (5) is sandwiched between two elastic bars (4 and 6) of similar cross-sectional area and Young
s modulus. The test chamber pressure of the Hopkinson bar varied from 400 to 600 kPa, results in strain rates of between 200 and 900 s
1. An elastic stress pulse is imparted into the incident bar (4) by impacting it with a striker bar (2). The impact of the striker bar generates a longitudinal compressive incident stress pulse, ri, equal to twice the length of the striker bar, that travels down the incident bar and is recorded by the strain gauge (3i) at the incident bar. The pulse then reaches the incident bar/specimen interface where part of the pulse is reflected in the form of a tensile stress Fig. 3. Schematic diagram of a compressive split Hopkinson pressure bar [23]. Fig. 2. Geometry and loading direction for specimens. R.O. Ochola et al. / Composite Structures 63 (2004) 455–467 459

R.O. Ochola et al. Composite Structures 63(2004)455-467 pulse, designated or, which is also recorded by the strain Based on the initial calibration factor, a refined cali gauge (3r)at the incident bar. The remainder of the bration factor is determined which accounts more ac stress pulse propagates through the specimen where curately for the specifics of each test and the data the specimen absorbs part of the energy of the pulse. recorded. The strain in the bars, obtained from applying The remaining pulse, ot, propagates along the trans- the refined calibration factor, is converted to stress by mitter bar, and is recorded by the strain gauge at the multiplying the strain with the Young,s modulus, E, transmitter bar, (3, according to Hooke's law, o= Ee. Therefore, for a given striker length, the strain rate As the elastic pulses are measured at the strain gauge can be increased by augmenting the impact velocity of locations in a particular distance from the specimen, it is he striker. Increasing the length of striker, whilst necessary to"shift"these signals to the bar/specimen keeping the impact velocity constant increases the strain interface in order to obtain information on the elastic that the specimen undergoes. The length of the striker pulses in the specimen. A longitudinal elastic wave that must always remain less than half the length of the propagates through a round bar is subjected to disper hortest pressure bar. This ensures that no overlap of the sion. The effect of dispersion, namely higher frequency incident pulse and the reflected pulse occurs at the strain components of the wave travel with a lower velocity gauges (3; D) Eqs.(1)(12) constitute the full set of equations nec- in the"shifting "procedure. The shifting of the pulse sary for post processing of the data captured by the achieved by"undispersing"the reflected and transmit strain gauges on incident bar and transmitter bar. The ted pulses, as well as"dispersing"the incident pulse, bars should remain elastic during the test. The maxi- shown in Fig. 5(a), as elucidated by Love [21]. If the mum allowable striker velocity, Umax, can be calculated cross-sectional area of the specimen differs from that of from Youngs modulus(E), and the yield strength(oy) of the bar, Eq (10)is employed to calculate the stress in the he incident and transmitter bar using Eq(13) cOdy Incident pulse Co is the longitudinal wave velocity. From the three sets of strain gauge readings viz. incident pulse, reflected lse, and transmitted pulse the time dependent stress 50 Transmitted pulse tate in the bar is determined using one-dimensiona wave propagation theory 3.4. Hopkinson bar data processing The signals of the strain gauges on the incident bar and the transmitter bar of the ShPB as voltage readings as illustrated in Fig. 4. The voltage Time(micro sec) readings of the incident, reflected and transmitted elastic pulse, depicted in Fig. 4, are converted to strain data using an initial calibration factor which relates the voltage signal to the actual strain at the strain gauge 150 0000020.040.06 Fig. 5.(a) Test data showing the incident and reflected pulse dispersed 0.0005 0.0015 to the bar/specimen interface and the transmitted pulse undispersed to the specimen/bar interface.(b) Stress-strain and strain plot for speci- men as converted from test data given in(a) from strain gauges on the Fig. 4. Raw data from strain gauges on incident and transmitter bars incident and transmitter bars

pulse, designated rr, which is also recorded by the strain gauge (3r) at the incident bar. The remainder of the stress pulse propagates through the specimen where the specimen absorbs part of the energy of the pulse. The remaining pulse, rt, propagates along the trans￾mitter bar, and is recorded by the strain gauge at the transmitter bar, (3t). Therefore, for a given striker length, the strain rate can be increased by augmenting the impact velocity of the striker. Increasing the length of striker, whilst keeping the impact velocity constant increases the strain that the specimen undergoes. The length of the striker must always remain less than half the length of the shortest pressure bar. This ensures that no overlap of the incident pulse and the reflected pulse occurs at the strain gauges (3i; t). Eqs. (1)–(12) constitute the full set of equations nec￾essary for post processing of the data captured by the strain gauges on incident bar and transmitter bar. The bars should remain elastic during the test. The maxi￾mum allowable striker velocity, vmax, can be calculated from Young
s modulus (E), and the yield strength (ry) of the incident and transmitter bar using Eq. (13) vmax ¼ 2C0ry E ð13Þ C0 is the longitudinal wave velocity. From the three sets of strain gauge readings viz. incident pulse, reflected pulse, and transmitted pulse the time dependent stress state in the bar is determined using one-dimensional wave propagation theory. 3.4. Hopkinson bar data processing The signals of the strain gauges on the incident bar and the transmitter bar of the SHPB set-up are recorded as voltage readings as illustrated in Fig. 4. The voltage readings of the incident, reflected and transmitted elastic pulse, depicted in Fig. 4, are converted to strain data using an initial calibration factor which relates the voltage signal to the actual strain at the strain gauge. Based on the initial calibration factor, a refined cali￾bration factor is determined which accounts more ac￾curately for the specifics of each test and the data recorded. The strain in the bars, obtained from applying the refined calibration factor, is converted to stress by multiplying the strain with the Young
s modulus, E, according to Hooke
s law, r ¼ Ee. As the elastic pulses are measured at the strain gauge locations in a particular distance from the specimen, it is necessary to ‘‘shift’’ these signals to the bar/specimen interface in order to obtain information on the elastic pulses in the specimen. A longitudinal elastic wave that propagates through a round bar is subjected to disper￾sion. The effect of dispersion, namely higher frequency components of the wave travel with a lower velocity than lower frequency components, has to be considered in the ‘‘shifting’’ procedure. The shifting of the pulse is achieved by ‘‘undispersing’’ the reflected and transmit￾ted pulses, as well as ‘‘dispersing’’ the incident pulse, shown in Fig. 5(a), as elucidated by Love [21]. If the cross-sectional area of the specimen differs from that of the bar, Eq. (10) is employed to calculate the stress in the -5 -2.5 0 2.5 5 0 0.0005 0.001 0.0015 Time (s) Voltage (V) Fig. 4. Raw data from strain gauges on incident and transmitter bars. -100 (a) (b) -50 0 50 100 150 250 350 Time (micro sec) Stress (MPa) 0 50 100 150 200 250 300 350 400 450 0.00 0.02 0.04 0.06 0.08 Strain Stress (MPa) 0 100 200 300 400 500 600 700 800 900 Strain rate (s-1) Reflected pulse Transmitted pulse Incident pulse Fig. 5. (a) Test data showing the incident and reflected pulse dispersed to the bar/specimen interface and the transmitted pulse undispersed to the specimen/bar interface. (b) Stress–strain and strain plot for speci￾men as converted from test data given in (a) from strain gauges on the incident and transmitter bars. 460 R.O. Ochola et al. / Composite Structures 63 (2004) 455–467

RO. Ochola et al. Composite Structures 63 (2004)455-467 specimen from the stress in the bar at the specimen/bar interface. The strain rate is calculated where the velocity of the transmit se in the bar. u 00 with a is the transmitted stress pulse in at the bar/ specimen interface calculated as described in this sec- ion. The strain in the specimen is calculated by inte- Strain grating the strain rate formulated in Eq. (14) increase in strain to a maximum of 400 mpa and de- creases thereafter. The strain rate increases with in creasing strain to 800 s-I after which it decreases rapidly. The ratio of cross-sectional area of the bar that of the specimen was 10: 1. Fig. 5(a) indicates a maximum stress, at, in transmitter bar of approximately 41 MPa. Applying Eq. (9)and considering that 0i-Or= ot results in a maximum stress in the specimen 398 MPa. This value concurs very well with the peak 350 value of the stress shown in Fig. 5(b) 4. Statistical analysis of data Due to the manufacturing techniques of the polymer Fig. 7. Stress vs standard deviation for CFRP and GFRP at quas matrix laminate specimens tested, coupled with the brittle nature of failure modes that include fibre failure matrix cracking and delamination, invariably the strength of these orthotropic materials are expected to vary slightly. Statistical analysis has been employed to ensure a certain degree of accuracy in the experimental data. However, it is suggested that no quantitative method exists to determine what degree of accuracy is sufficient for measuring the compressive stress values for brittle laminate polymer composite materials A series of tests done on CFRP and GFRP at quasi tatic and dynamic loading resulted in stress vs strain data presented as a graph in Figs. 6-8, respectively. The maximum stress values of the stress strain curves shown indicate the ultimate stress for the materials. The results 0.000010.020.030.040.050.06 Strain of the quasi-static and dynamic tests on sets of 15 specimens of CFRP and GFRP, together with the mean Fig. 8. Comparison of stress values for CFRP and GFRP at dynam values and standard deviation of the mean(SD), are strain rates 400 presented in Tables I and 3, respectively The reproducibility of the ultimate stress results ob- the expected compressive strength (expected)using Eq ained from both low and high strain rate tests are de- (16)in order to determine the acceptability of the ex- pendent on the degree of homogeneity of the composite pected value(Expected) material. The mean compressive strength value(X)and the standard deviation of the mean( SD) is compared to Expected=r+SDI

specimen from the stress in the bar at the specimen/bar interface. The strain rate is calculated using e_ ¼ v l0ðtÞ ð14Þ where the velocity of the transmitted pulse in the bar, v, is v ¼ rt qbC0;b ð15Þ with rt is the transmitted stress pulse in at the bar/ specimen interface calculated as described in this sec￾tion. The strain in the specimen is calculated by inte￾grating the strain rate formulated in Eq. (14). Fig. 5(b) depicts the stress and strain rate vs. strain in the specimen converted from results shown in Fig. 5(a). The stress in the specimen is shown to increase with an increase in strain to a maximum of 400 MPa and de￾creases thereafter. The strain rate increases with in￾creasing strain to 800 s
1 after which it decreases rapidly. The ratio of cross-sectional area of the bar to that of the specimen was 10:1. Fig. 5(a) indicates a maximum stress, rt, in transmitter bar of approximately 41 MPa. Applying Eq. (9) and considering that ri
rr ¼ rt results in a maximum stress in the specimen of 398 MPa. This value concurs very well with the peak value of the stress shown in Fig. 5(b). 4. Statistical analysis of data Due to the manufacturing techniques of the polymer matrix laminate specimens tested, coupled with the brittle nature of failure modes that include fibre failure, matrix cracking and delamination, invariably the strength of these orthotropic materials are expected to vary slightly. Statistical analysis has been employed to ensure a certain degree of accuracy in the experimental data. However, it is suggested that no quantitative method exists to determine what degree of accuracy is sufficient for measuring the compressive stress values for brittle laminate polymer composite materials. A series of tests done on CFRP and GFRP at quasi￾static and dynamic loading resulted in stress vs. strain data presented as a graph in Figs. 6–8, respectively. The maximum stress values of the stress strain curves shown indicate the ultimate stress for the materials. The results of the quasi-static and dynamic tests on sets of 15 specimens of CFRP and GFRP, together with the mean values and standard deviation of the mean (SD), are presented in Tables 1 and 3, respectively. The reproducibility of the ultimate stress results ob￾tained from both low and high strain rate tests are de￾pendent on the degree of homogeneity of the composite material. The mean compressive strength value (X) and the standard deviation of the mean (SD) is compared to the expected compressive strength (Xexpected) using Eq. (16) in order to determine the acceptability of the ex￾pected value (Xexpected). Xexpected ¼ jX SDj ð16Þ 0 100 200 300 400 500 600 0 0.05 0.1 0.15 Strain Stress (MPa) Fig. 6. Stress vs. strain for CFRP and GFRP at quasi-static strain rates. 200 250 300 350 400 450 500 550 600 650 0123 Standard deviation range Stress (MPa) GFRP CFRP Fig. 7. Stress vs. standard deviation for CFRP and GFRP at quasi￾static strain rates. 0 100 200 300 400 500 600 700 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Strain Stress (MPa) Fig. 8. Comparison of stress values for CFRP and GFRP at dynamic strain rates 400 s
1. R.O. Ochola et al. / Composite Structures 63 (2004) 455–467 461

462 RO. Ochola et al. Composite Structures 63(2004)455-467 Table 1 Stress data for CFRP and GFRP tested at quasi-static strain rates LXbest -expected No of test Strain rate CFRP stress GFRP stress (MPa) An example of the statistical analysis performed is il- 456 000000 lustrated using the data in Table 1. At quasi-static strain rates the best value, Xbest, from Table l for the CFRP 333 405 specimen is equal to 493 MPa and the mean value Expected =X =495 MPa with a standard deviation (SD) of 49. Using Eq(17) the number of standard deviation is calculated to be s=0.04. The probability of obtaining a stress value that differs from Expected by s or more 10 000000000 standard deviations is determined from Eq (18) 497 P(outside t(SD))=l-P(within t(SD)) 13 P(outside t(SD))=0.96 Using the table of normal error integral [28] and under Mean the assumption of complete homogeneity of the CFRP pecimens, from the Gaussian distribution of the test data, t=0.04 is equivalent to a probability of 66.3% hat the next strength value for CFRP will fall within one standard deviation of the mean The worst value in Table 2 Standard deviation of compressive stress mean values for CFRP and Table I is 384 MPa(from test no. 1)which lies outside GFRP at quasi-static strain rate he limit of one standard deviation of the mean. viz. 495 Number (n) (X-n(SD) MPa. The same statistical analysis is carried out on the GFRP system at low strain rates, as well as both CFRP CFRP)2 and GFRP systems at high strain rates using Eqs.(16)- (18), respectively The results of the statistical analysis show the prob- ability that the next quasi-static compressive strength value will lie within one standard deviation of the mean Table 3 for GFRP at low strain rates is 66.8%(Fig. 9). The Stress data for CFRP and GFRP tested at dynamic strain rates probability that the next dynamic compressive strength CFRP stress Strain GFRP stress value for cfrp and gfrp will lie within one standard deviation of the mean is 59. 4% and 62.1%o, respectively In this study, therefore, an accuracy of one standard deviations of the mean value is considered to be suffi- cient to determine whether the materials tested may be 450 450 526 00000000 42098 忘400 200 In order to dete whether or not the next expected 450 value is acceptable, the following statistical analysis is Strain rate(s") performed. The number of standard deviation, s, by Fig. 9. Mean stress to one standard deviation vs. strain rate plot for which Xbest differs from Expected is given in Eq(17) GFRP system

In order to determine whether or not the next expected value is acceptable, the following statistical analysis is performed. The number of standard deviation, s, by which Xbest differs from Xexpected is given in Eq. (17) s ¼ jXbest
Xexpectedj SD  ð17Þ An example of the statistical analysis performed is il￾lustrated using the data in Table 1. At quasi-static strain rates the best value, Xbest, from Table 1 for the CFRP specimen is equal to 493 MPa and the mean value Xexpected  X  495 MPa with a standard deviation (SD) of 49. Using Eq. (17) the number of standard deviation is calculated to be s ¼ 0:04. The probability of obtaining a stress value that differs from Xexpected by s or more standard deviations is determined from Eq. (18). Pðoutside tðSDÞÞ ¼ 1
Pðwithin tðSDÞÞ Pðoutside tðSDÞÞ ¼ 0:96 ð18Þ Using the table of normal error integral [28] and under the assumption of complete homogeneity of the CFRP specimens, from the Gaussian distribution of the test data, t ¼ 0:04 is equivalent to a probability of 66.3% that the next strength value for CFRP will fall within one standard deviation of the mean. The worst value in Table 1 is 384 MPa (from test no. 1) which lies outside the limit of one standard deviation of the mean, viz. 495 MPa. The same statistical analysis is carried out on the GFRP system at low strain rates, as well as both CFRP and GFRP systems at high strain rates using Eqs. (16)– (18), respectively. The results of the statistical analysis show the prob￾ability that the next quasi-static compressive strength value will lie within one standard deviation of the mean for GFRP at low strain rates is 66.8% (Fig. 9). The probability that the next dynamic compressive strength value for CFRP and GFRP will lie within one standard deviation of the mean is 59.4% and 62.1%, respectively. In this study, therefore, an accuracy of one standard deviations of the mean value is considered to be suffi- cient to determine whether the materials tested may be Table 1 Stress data for CFRP and GFRP tested at quasi-static strain rates No. of test Strain rate CFRP stress (MPa) GFRP stress (MPa) 1 10
3 384 520 2 10
3 502 *554 3 10
3 397 507 4 10
3 476 490 5 10
3 542 551 6 10
3 520 432 7 10
3 499 551 *8 *10
3 *566 476 9 10
3 522 501 10 10
3 469 499 11 10
3 487 516 12 10
3 490 497 13 10
3 530 504 14 10
3 493 478 15 10
3 550 547 Mean 495 508 SD 49 33 Stress. Table 3 Stress data for CFRP and GFRP tested at dynamic strain rates No. of test Strain rate CFRP stress (MPa) Strain rate GFRP stress (MPa) 1 450 433 450 594 *2 450 521 450 650 3 450 467 450 563 4 450 532 450 508 5 450 515 450 591 6 450 526 450 596 7 450 490 450 632 8 450 454 450 599 9 450 522 450 622 10 450 503 450 637 11 450 459 450 605 12 450 487 450 596 13 450 524 450 579 14 450 510 450 612 15 450 523 450 645 Mean 498 602 SD 30 35 Table 2 Standard deviation of compressive stress mean values for CFRP and GFRP at quasi-static strain rate Number (n) (X
nðSDÞ) (X) (X þ nðSDÞ) (CFRP) 1 446 495 544 (CFRP) 2 397 495 593 (GFRP) 1 475 508 541 (GFRP) 2 442 508 574 0 100 200 300 400 500 600 700 0.001 450 Strain rate (s-1) stress (MPa) Fig. 9. Mean stress to one standard deviation vs. strain rate plot for GFRP system. 462 R.O. Ochola et al. / Composite Structures 63 (2004) 455–467

RO. Ochola et al. Composite Structures 63 (2004)455-467 GFRE CFRP 500 g400 200 0 Fig. 10. Stress vs standard deviation for CFRP and GFRP at d Strain Fig. Il. Stress-strain plot for CFRP and GfRP tested at varying deemed strain rate sensitive. From Fig. 7 no significant strain rates. improvement in the nature of scatter of the data is note by considering two standard deviation of the mean. Therefore any constitutive relation proposed will con- For improved impact resistance, a large strain to sider the data at one standard deviation of the mean failure is desirable. This suggests that the impact resis- (Fig.10) tance of gfrP reduces with an increase in strain rate. It appears that for compression loading the CFrP is more strain rate sensitive than the gfrp as far as strain to failure is concerned. this dis 5. Results and discussion what is reported in literature, namely that CFRP sys tems are strain rate insensitive, or considerably less The stress-strain behaviour of CFRP at varying strain rate sensitive than GFRP systems. It should strain rates is seen to vary with respect to the strain to however, be noted that the results reported in literature failure experienced by the test specimens. The ratio of refer mainly to tensile loading and ultimate stress (27 the strain to failure for CFRP at low to high strain rate while here the suggestion of strain rate sensitivity of is 9%/2%=4.5 Gilat et al. [24]reports the same strain to CfrP is based on strain to failure under compressive failure of 2% at dynamic strain rates. It is suggested loading therefore that at lower loading rates the laminate spe A subsequent study also investigates the effects of imens has more time to distribute the load and under- strain rate on the compressive ultimate stress of both goes steady deformation. Higher strain to failure values composite systems. The mean values for the compressive are usually associated with larger energy absorption ultimate stress at quasi-static strain rate are given values which are represented by the area under the Table 2. At low strain rates CFRP and GFRP appear to stress-strain graph(Fig. I1). It is proposed by Cantwell have similar ultimate stress. viz. 495 and 508 MPa re- and Morton [26] that for improved impact resistance, spectively(see also Table 4) large strain to failure is desirable. This suggests that the This suggests that the type of fibre in the system has a impact resistance of CFRP under compression loading limited effect on the ultimate stress of the specimen un decreases with an increase in strain rate dergoing uniaxial compression in the fibre direction, In the case of the GFrP system at lower strain rates the strain to failure observed in Fig. 6(Glow)is 15%. The suggestive of matrix dominated behaviour. However, the results may also imply that carbon and glass fibre overall strain recorded for GFRP at quasi-static strain rates shows that the deformation of the cylindrical specimen is relatively uniform up to failure. At lower Table 4 strain rates the GFRP system has time to distribute the Standard deviation of compressive stress mean values for CFRP and load, hence the failure of the specimen takes place GFRP at dynamic strain rates radually. At the strain rate of 450 s- the strain (r-n(SD)) (X) Cr +n(S)) failure is 4%, from the curve Ghigh in Fig. 11. It is sug (CFRP)I gested that due to the rapid loading, the material tends a more brittle behaviour resulting in a low strain to (GFRP)I failure value (GFRP)2 672

deemed strain rate sensitive. From Fig. 7 no significant improvement in the nature of scatter of the data is noted by considering two standard deviation of the mean. Therefore any constitutive relation proposed will con￾sider the data at one standard deviation of the mean (Fig. 10). 5. Results and discussion The stress–strain behaviour of CFRP at varying strain rates is seen to vary with respect to the strain to failure experienced by the test specimens. The ratio of the strain to failure for CFRP at low to high strain rate is 9%/2% ¼ 4.5. Gilat et al. [24] reports the same strain to failure of 2% at dynamic strain rates. It is suggested therefore that at lower loading rates the laminate spec￾imens has more time to distribute the load and under￾goes steady deformation. Higher strain to failure values are usually associated with larger energy absorption values which are represented by the area under the stress–strain graph (Fig. 11). It is proposed by Cantwell and Morton [26] that for improved impact resistance, a large strain to failure is desirable. This suggests that the impact resistance of CFRP under compression loading decreases with an increase in strain rate. In the case of the GFRP system at lower strain rates, the strain to failure observed in Fig. 6 (Glow) is 15%. The overall strain recorded for GFRP at quasi-static strain rates shows that the deformation of the cylindrical specimen is relatively uniform up to failure. At lower strain rates the GFRP system has time to distribute the load, hence the failure of the specimen takes place gradually. At the strain rate of 450 s
1 the strain to failure is 4%, from the curve Ghigh in Fig. 11. It is sug￾gested that due to the rapid loading, the material tends to a more brittle behaviour resulting in a low strain to failure value. For improved impact resistance, a large strain to failure is desirable. This suggests that the impact resis￾tance of GFRP reduces with an increase in strain rate. It appears that for compression loading the CFRP is more strain rate sensitive than the GFRP as far as strain to failure is concerned. This disagrees to some extent with what is reported in literature, namely that CFRP sys￾tems are strain rate insensitive, or considerably less strain rate sensitive than GFRP systems. It should, however, be noted that the results reported in literature refer mainly to tensile loading and ultimate stress [27] while here the suggestion of strain rate sensitivity of CFRP is based on strain to failure under compressive loading. A subsequent study also investigates the effects of strain rate on the compressive ultimate stress of both composite systems. The mean values for the compressive ultimate stress at quasi-static strain rate are given in Table 2. At low strain rates CFRP and GFRP appear to have similar ultimate stress, viz. 495 and 508 MPa re￾spectively (see also Table 4). This suggests that the type of fibre in the system has a limited effect on the ultimate stress of the specimen un￾dergoing uniaxial compression in the fibre direction, suggestive of matrix dominated behaviour. However, the results may also imply that carbon and glass fibre 200 300 400 500 600 700 0123 Standard deviation range Stress (MPa) CFRP GFRP Fig. 10. Stress vs. standard deviation for CFRP and GFRP at dy￾namic strain rates. Table 4 Standard deviation of compressive stress mean values for CFRP and GFRP at dynamic strain rates Number (n) (X
nðSDÞ) (X) (X þ nðSDÞ) (CFRP) 1 468 498 528 (CFRP) 2 438 498 558 (GFRP) 1 567 602 637 (GFRP) 2 532 602 672 0 100 200 300 400 500 600 700 0.00 0.05 0.10 0.15 Strain Stress Ghigh Clow Glow CHigh (MPa) Fig. 11. Stress–strain plot for CFRP and GFRP tested at varying strain rates. R.O. Ochola et al. / Composite Structures 63 (2004) 455–467 463

RO. Ochola et al. Composite Structures 63(2004)455-467 have similar compressive ultimate stress at low strain rates, this trend is reported by Matthews and Rawlings [25] who gives the quasi-static longitudinal compression strength of glass and carbon to vary from 650-950 and 700-1200 MPa, respectively. At high strain rates, how- Direction of shear fracture ever,a clear discrepancy is noted between the ultimate stresses of cfrP and gfrP of 498 and 602 MPa. re- pectively. The mean compressive ultimate stress value for GFRP increases by 104 MPa(20.9%)as the strain rate is increased from 10- to 450 s-, this suggest the existence of a strengthening mechanism in the GFRP system. For CfRP the ultimate stress appears to be constant with a negligible increase of 3 MPa(0.6%)from 495 to 498 MPa as the strain rate is increased from low to high Fig. 12. Shear fracture from quasi-statically loaded CFRP show As strain to failure and ultimate stress for both sys- direction of shear fracture propagation. tems have been discussed further information is ex- pected from the comparison of the absorbed energies The stress strain curve for the CFRP system at both strain rates is illustrated in Fig. ll. The energy absorbed from the specimen is represented by the area under the curve. The observed energy is calculated to be: CFRP at low strain rate 17.1 J, CFRP at high strain rate 209J, GFRP at low strain rate 4.9 J, GFRP at high strain rate It can be seen that for low strain rates the energy absorbed (integral of ultimate stress as a function of strain to failure) by the gFRP system, 20.9 J is mar- ginally larger than the energy absorbed by the CFRP, 17.1 J. At high loading rates the energy absorbed by the GFRP(4.9 D)is again marginally higher than the energy absorbed by the CFRP(2.6 D) Strain rate(s") Despite the observed increase in ultimate stres pled with the decrease in strain to failure as the strain CFRP system Fig. 13. Mean stress to one standard deviation vs strain rate plot for rate is increased, an overall decrease in the energy ab- sorbed for CFrP and gfrP of 15% and 23.5%.re- spectively, of the high strain rate values is noted tensile stresses developed due to a mismatch of Poisson In order to elucidate the observed trends reported in ratio between the matrix and fibre. For CFRP at high strain to failure and compressive ultimate stress, the strain rates, clear separation at laminar interfaces cou failure modes are investigated. It is hoped that the pled with fibre and matrix disintegration was observed changes in failure modes would account for the varia- by visual inspection. The reason for this failure mecha- tion in the material properties. At low strain rates nism for CFRP at high strain rates is not well under- (quasi-static) the dominant damage mode for CFRP was stood and is also noted by Hosur et al. [20]. More work shear failure, Fig. 12. This failure mechanism is typical therefore required to explain the main reason why the for brittle materials. The compression load is transferred different modes of failure examined in the CFRP system at steady state along the fibres with the fibres failing at low and high strain rates results in similar ultimate along an inclined shear fracture plane due to the con- failure stresses as described earlier and indicated in Fig centration of the compression load in the fibre. The 1 main modes of failure observed for CFRP at high strain In the case of GfRP the failure sequence of fibre rates are delamination, fibre/matrix disintegration and inking, micro-buckling followed by shear failure, was debonding with very little fibre buckling, kinking or found at low strain rates, with the failure modes fibre breakage. Debonding occurs as a result of failure changing to longitudinal splitting and delamination initiated at a point in the fibre/matrix interface which the strain rate is increased to 450 s for GFRP(Fig then propagates through the entire length of the inter- 14). The change in failure modes results in an increase in face. It usually occurs when the interfacial stress exceeds the compressive strength from quasi-static to dynamic the interfacial strength, this results from strain rates

have similar compressive ultimate stress at low strain rates, this trend is reported by Matthews and Rawlings [25] who gives the quasi-static longitudinal compression strength of glass and carbon to vary from 650–950 and 700–1200 MPa, respectively. At high strain rates, how￾ever, a clear discrepancy is noted between the ultimate stresses of CFRP and GFRP of 498 and 602 MPa, re￾spectively. The mean compressive ultimate stress value for GFRP increases by 104 MPa (20.9%) as the strain rate is increased from 10
3 to 450 s
1, this suggest the existence of a strengthening mechanism in the GFRP system. For CFRP the ultimate stress appears to be constant with a negligible increase of 3 MPa (0.6%) from 495 to 498 MPa as the strain rate is increased from low to high, respectively. As strain to failure and ultimate stress for both sys￾tems have been discussed further information is ex￾pected from the comparison of the absorbed energies. The stress strain curve for the CFRP system at both strain rates is illustrated in Fig. 11. The energy absorbed from the specimen is represented by the area under the curve. The observed energy is calculated to be: CFRP at low strain rate 17.1 J, CFRP at high strain rate 20.9 J, GFRP at low strain rate 4.9 J, GFRP at high strain rate 2.6 J. It can be seen that for low strain rates the energy absorbed (integral of ultimate stress as a function of strain to failure) by the GFRP system, 20.9 J is mar￾ginally larger than the energy absorbed by the CFRP, 17.1 J. At high loading rates the energy absorbed by the GFRP (4.9 J) is again marginally higher than the energy absorbed by the CFRP (2.6 J). Despite the observed increase in ultimate stress cou￾pled with the decrease in strain to failure as the strain rate is increased, an overall decrease in the energy ab￾sorbed for CFRP and GFRP of 15% and 23.5%, re￾spectively, of the high strain rate values is noted. In order to elucidate the observed trends reported in strain to failure and compressive ultimate stress, the failure modes are investigated. It is hoped that the changes in failure modes would account for the varia￾tion in the material properties. At low strain rates (quasi-static) the dominant damage mode for CFRP was shear failure, Fig. 12. This failure mechanism is typical for brittle materials. The compression load is transferred at steady state along the fibres with the fibres failing along an inclined shear fracture plane due to the con￾centration of the compression load in the fibre. The main modes of failure observed for CFRP at high strain rates are delamination, fibre/matrix disintegration and debonding with very little fibre buckling, kinking or fibre breakage. Debonding occurs as a result of failure initiated at a point in the fibre/matrix interface which then propagates through the entire length of the inter￾face. It usually occurs when the interfacial stress exceeds the interfacial strength, this results from transverse tensile stresses developed due to a mismatch of Poisson
s ratio between the matrix and fibre. For CFRP at high strain rates, clear separation at laminar interfaces cou￾pled with fibre and matrix disintegration was observed by visual inspection. The reason for this failure mecha￾nism for CFRP at high strain rates is not well under￾stood and is also noted by Hosur et al. [20]. More work is therefore required to explain the main reason why the different modes of failure examined in the CFRP system at low and high strain rates results in similar ultimate failure stresses as described earlier and indicated in Fig. 13. In the case of GFRP the failure sequence of fibre kinking, micro-buckling followed by shear failure, was found at low strain rates, with the failure modes changing to longitudinal splitting and delamination as the strain rate is increased to 450 s
1 for GFRP (Fig. 14). The change in failure modes results in an increase in the compressive strength from quasi-static to dynamic strain rates. Fig. 12. Shear fracture from quasi-statically loaded CFRP showing direction of shear fracture propagation. 0 100 200 300 400 500 600 0.001 450 Strain rate (s-1) Stress (MPa) Fig. 13. Mean stress to one standard deviation vs. strain rate plot for CFRP system. 464 R.O. Ochola et al. / Composite Structures 63 (2004) 455–467