Further development of a data reduction method for the nonlinear viscoelastic characterization of FRPs

Composites: Part A 30 (1999) 839–848

Further development of a data reduction method for the nonlinear viscoelastic characterization of FRPs G.C. Papanicolaou a,*, S.P. Zaoutsos a, A.H. Cardon b a

b

Composite Materials Group, Department of Mechanical and Aeronautical Engineering, University of Patras, 265 00 Patras, Greece Composite Systems and Adhesion Research Group, Department of Mechanics of Materials and Constructions, Free University of Brussels (VUB), B-1050 Brussels, Belgium Received 5 December 1997; received in revised form 20 November 1998; accepted 5 January 1999

Abstract According to the well-known Schapery’s formulation, the nonlinear viscoelastic response of any material is controlled by four stress and temperature dependent parameters, g0 g1, g2 and as , which reflect the deviation from the linear viscoelastic response. Based on Schapery’s formulation, a new methodology for the separate estimation of the three out of four nonlinear viscoelastic parameters, g0, g1 and as , was recently developed by the authors. In the present article, a further development of the previously developed methodology is attempted leading to an analytical estimation of the fourth nonlinear parameter, g2, which additionally includes the viscoplastic response of the system. Thus, a full nonlinear characterization of the composite system under consideration is achieved. The validity of the integrated model was verified through creep-recovery experiments, applied at different stress levels to a unidirectional carbon fibre reinforced polymer. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: Nonlinear behaviour; B. Creep

1. Introduction The long-term mechanical characterization of fibre reinforced plastics has recently received much attention, as the use of these materials for manufacturing structural parts rapidly increases in a wide spectrum of engineering applications. Owing to the polymeric matrix viscoelastic response revealed under mechanical and environmental conditions, an important part of the long-term mechanical behaviour of polymer based composites is also viscoelastic. This behaviour becomes more complex at high loading levels where the behaviour becomes nonlinear. Study of the nonlinear viscoelastic behaviour led to a number of constitutive equations most of them describing not only the time dependency of the material under consideration, but also the effect of nonlinearity introduced because of the high stress levels applied. Early attempts to describe the nonlinear viscoelastic response, led to the Modified Superposition Principle (MSP) which was proposed by Leaderman [1] and was also used by Findley et al. [2]. A most common formulation, quite applicable and * Corresponding author.

simple in use, was introduced by Schapery [3,4]. According to this model the nonlinearity is controlled by four stress and temperature dependent parameters, g0, g1, g2 and as , which reflect the deviation from the linear viscoelastic response. Lou and Schapery [5] proposed a mixed graphical and numerical technique for the estimation of these parameters using one-step creep-recovery experiments. Schapery’s formulation was used by various researchers [6–8] for the description of the viscoelastic response of many polymeric systems. It was also proved adequate for describing complex loading histories [9]. In a recent work [10], the authors developed a new methodology where three out of the four Schapery nonlinear parameters were separately determined from simple step creep-recovery curves. More precisely, both limiting experimental values resulted from a simple creep-recovery test at a reference stress level and analytical expressions arising from Schapery’s constitutive equation for the specific test, are utilized for the analytical determination of g0 and g1, while as was numerically determined from the recovery data. In the present work a further development of the earliermentioned methodology is presented through the analytical estimation of the nonlinear parameter g2.

1359-835X/99/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S1359-835 X( 99)00 004-4

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Fig. 1. A typical strain vs. time curve, for a creep/recovery test.

2. Theoretical background 2.1. Linear and nonlinear constitutive modelling principles Boltzmann, based on the linear stress–strain behaviour, suggested the following formulation for the viscoelastic strain response 1 (t), of a material under a constant stress s 0, applied at t ˆ 0: Zt ds 1…t† ˆ D0 s0 1 DD…t 2 t† dt ; …1† dt 0 where D0 is the initial, time-independent compliance component and DD(t) is the transient, time-dependent compliance component. The integral appearing in Eq. (1) is called the hereditary integral and this expression shows that the strain at any given time depends on the entire stress history. The single integral constitutive equation, developed by Schapery [3–5], for the description of the nonlinear viscoelastic behaviour under isothermal and uniaxial loading conditions, can be given in the following form: Zt d…g2 s† 1…t† ˆ g0 D0 s0 1 g1 DD…c 2 c 0 † dt ; …2† dt 0 where DD(c ) is the transient, time-dependent compliance component, while c and c 0 are the so-called reduced times defined as: Zt dt 0 Zt dt 0 cˆ and c 0 ˆ c…t† ˆ 0 as 0 as and g0, g1, g2, are the stress dependent nonlinear material’s parameters.

Each of these parameters defines a nonlinear effect on the compliance of the material. The factor g0 defines stress and temperature effects on the instantaneous elastic compliance and is a measure of the state dependent stiffness variation. Transient compliance factor g1 has a similar meaning, operating on the creep compliance component. Factor g2 accounts for the influence of the loading rate on creep, and depends on stress and temperature. The factor as is a time scale shift factor. The latter factor is in general a stress and temperature dependent function and modifies the viscoelastic response as a function of temperature and stress. Mathematically as , shifts the creep data parallel to the time axis relative to a master curve for creep strain vs. time. Schapery’s constitutive equation also includes the linear case where g0 ˆ g1 ˆ g2 ˆ as ˆ 1 and leads to the Boltzmann’s Superposition Principle described by Eq. (1). By introducing an additional term, e vp(t), accounting for the time dependent viscoplastic response of the material, Eq. (2) takes the form:

1…t† ˆ g0 D0 s0 1 g1

Zt 0

DD…c 2 c 0 †

d…g2 s† dt 1 1vp …t†: …3† dt

Referring to Fig. 1, for a creep-recovery test, where a constant stress s 0 is applied at t ˆ 0 and subsequently removed at t ˆ ta, the stress history can be described as:

s…t† ˆ s0 ‰H…t† 2 H…t 2 ta †Š;

…4†

where H(t) is the Heaviside step function. Substitution of the stress history, as described by Eq. (4), into Schapery’s Eq. (3) results for the strain response of the

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creep and recovery phase, respectively:   t 1c …t† ˆ g0 D0 s0 1 g1 g2 DD s 1 1vp …t†; as 0

…5†

for 0 , t , ta     t 1r …t† ˆ g2 DD a 1 t 2 ta 2 DD…t 2 ta † s0 1 1vp …ta1 †; as for t . ta :

…6†

Assuming that the compliance function follows a power law of time, i.e. DD…t† ˆ Ctn , then (5) and Eq. (6) can be written as:  n t 1c …t† ˆ g0 D0 s0 1 g1 g2 C s0 1 1vp …t†; as …7† for 0 , t , ta and

  n  t 1r …t† ˆ C a 1 t 2 ta 2C…t 2 ta †n g2 s0 as

while for t ˆ ta1 ; Eq. (6) can be written as: ! ta1 1 g s 1 1vp …ta1 †: 1r …ta † ˆ DD as 2 0

…8†

For a perfect linear viscoelastic behaviour the instantaneous strain response at t ˆ 0 1 is equal to the instantaneous strain jump at t ˆ ta1 (Fig. 1). However, in the nonlinear case these two responses are not equal. This is a significant point which enables us to continue in a more detailed investigation of the momentary responses at the time of instantaneous loading and unloading, respectively. If D1 0 is the difference between the two responses corresponding to the time of the instantaneous unloading …t ˆ ta1 † and the time of the instantaneous loading (t ˆ 0 1), respectively, then we obtain:

 D1c  …1 1 as l†n 2 …as l†n 1 1vp …ta1 †; g1

…9†

for t . ta ; where l ˆ ……t† 2 …ta †=…ta ††, is the so-called nondimensional time and D1 c is the amount of transient strain accumulated during creep, given by:  n t …10† D1 c ˆ g 1 g 2 C a s 0 : as Eq. (9) also includes the linear case where g1 ˆ as ˆ 1. In this case, Eq. (9) becomes:

1r …t† ˆ D1c ‰…1 1 l†n 2 …l†n Š 1 1vp …ta1 †;

for t . ta ; …11†

1c …ta2 †

2

1r …ta1 †

ˆ

1c …ta2 †

! ta1 2 g2 DD s0 2 1vp …ta1 † as …17†

and 1 0c is given by Eq. (13) as:

10c ˆ g0 D0 s0 ˆ

1c …ta2 †



 ta2 2 g1 g2 DD s 2 1vp …ta2 †: …18† as 0

It is obvious of course that 1vp …ta2 † ˆ 1vp …ta1 † ˆ 1vp …ta †: Substitution of Eq. (17) and (18) into Eq. (16) yields to:   t …19† D10 ˆ g2 …g1 2 1†DD a s0 : as Partial division of Eq. (15) and (19) results in: g1 ˆ

D1c 2 1vp : D1c 2 D10 2 1vp

…20†

2.3. Numerical estimation of the stress shift factor as

where D1c ˆ Ctan s0 :

…16†

where e 0r is given by Eq. (14) as:

10r ˆ

or

…14†

However, the viscoelastic creep response at t ˆ ta2 can be found by subtracting the instantaneous elastic response, g0D0s 0, from the creep response given by Eq. (13), thus:  2 t D1c …ta2 † ˆ g1 g2 DD a s0 1 1vp …ta2 †: …15† as

D10 ˆ 10r 2 10c ;

11vp …ta1 †; for t . ta

1r …t† ˆ

841

…12†

2.2. Analytical estimation of the nonlinear parameter g1 According to the analytical method developed in a previous work [10], the estimation of the nonlinear parameter g1, can be achieved as follows: For t ˆ ta2 ; Eq. (5) can be written in the following form:  2 t 2 1c …ta † ˆ g0 D0 s0 1 g1 g2 DD a s0 1 1vp …ta2 †; …13† as

For the estimation of the stress factor as , the nondimensional formulation proposed by Lou and Schapery [5], and thoroughly investigated by Hiel [11] is used. According to the earlier method, having analytically determined the values of g0 and g1, as can be determined using Eq. (9). Especially in this case, as D1 c and 1 vp can also be experimentally measured, a proper value of as can be found by fitting the experimental values of recovery response 1 r(t) in Eq. (9). The only unknown parameter in Eq. (9) is the value of the n-exponent. This value can be easily calculated, by fitting the recovery data obtained from a creep-recovery

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test in the linear region, i.e. at a low stress level, to Eq. (11). 3. The proposed model for the estimation of the nonlinear parameter g2 In the present section, a new technique for the estimation of the nonlinear parameter g2 included in Schapery’s formulation is presented. The technique follows an analytical expression arising from Schapery’s viscoelastic model and Boltzmann’s principle as applied in the linear case. If we assume that the transient compliance DD(t) follows Findley’s power law then according to the analytical expression given by Eq. (19), the difference between the jumps at the time of loading and unloading of the system, can be written as:  n t …21† D10…nl† ˆ g2 …g1 2 1†C a s0…nl† : as It is also obvious that the pure viscoelastic response can be written from Eq. (12) as: D1c…l† ˆ Ctan s0…l† ;

…22†

where the indices l and nl denote magnitudes associating to linear and nonlinear viscoelastic response, respectively. By partial division of Eq. (21) and (22), we obtain: g2 ˆ

D10…nl† s0…l† ans : D1c…l† …g1 2 1† s0…nl†

…23†

It is very significant to note that the earlier mathematical treatment for the linear and the nonlinear viscoelastic response in both cases refers to the same time domain. This is the basic assumption of the model for its application to a series of creep-recovery tests at different stress levels. In summarizing the already described procedure we may present the following step listing for the nonlinear parameters evaluation: Step 1

Step 2

Step 3

Step 4

Step 5

g0 estimation is performed by comparing the instantaneous compliance value for the linear case with the respective instantaneous compliance value for the nonlinear case. From Eq. (20) the analytical estimation of g1 can be performed using appropriate creep-recovery experimental results. Eq. (11) is numerically curve fitted to the linear recovery data and the value of the n exponent is determined. Similarly, the numerical estimation of as can be performed by fitting the recovery data to the nondimensional formulation given by Eq. (9). Having known the values of as and g1, the value of g2 is analytically estimated using Eq. (23).

4. Experimental procedure The experimental results used for the evaluation of the present model are the same used by the authors in a recent work [10]. The manufacturing of the specimens as well as the experiments were conducted in the Department of Mechanics of Materials and Constructions of the Free University of Brussels (VUB). Creep-recovery data were obtained on a 908 carbon– epoxy composite. Standard prepreg types fibredux-920cts-5-42 with nominal weight 0.231 kg/m 2 were used for the fabrication of 12-layer carbon-epoxy unidirectional composite plates using the hand lay-up technique. All plates were cured in a scholtz autoclave following the standard curing cycle proposed by ciba-geigy consisted of a 300 kN/m 2 pressure and a curing temperature of 1258 for 120 min applied to the plates in the autoclave. Laminates were also postcured in an air oven at about 1208C for 3 h and then slowly cooled in the oven. Transverse straight-sided specimens with a nominal size 300 mm length, 17 mm width and 2 mm thickness, were cut from the unidirectional plates using a diamond wheel saw. End tabs were also glued to both sides of the edges of each specimen, using lexan tabbing material. The nominal size of each tab was 50 mm long, 17 mm wide and 2 mm thick. Two Micromeasurement ea-13-240lz-120 and cea-13240uz-120 strain gauges were mounted back to back on the center of each specimen along the loading direction using M-Bond 610 adhesive. Both short-term tensile elastic modulus and ultimate tensile strength tests were performed for five specimens using an Instron tensile testing machine. Strains were measured using electrical resistance foil gauges bonded to the specimens. A displacement rate of 1 mm/s was used for all the tests. The short-term material properties are listed in Table 1. The long term loading mode applied, was a simple tensile loading mode, consisted of an initial applied tensile loading, followed by an 168 h constant stress level, s 0, and an 168 h recovery. Six different stress levels, s 0, corresponding to 30%, 40%, 50%, 55%, 60% and 70% of the tensile rupture stress s u were applied. All the specimens were wired and set in the creep frame for at least 24 h prior to loading. All tests were performed using a four-station dead load creep frame. The weights used in each test for the loading of the specimens, were carefully calibrated before each testing. An oil pump was used to apply and remove the load in the Table 1 Short-term material data Tensile modulus Ultimate tensile strength Poisson’s ratio, y

E1 (longitudinal) ˆ 120 GPa E2 (transverse) ˆ 8.5 GPa (transverse) ˆ 65 MPa ˆ 0.34

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Fig. 2. Creep response of the 908 degrees carbon–epoxy composite for different applied tensile stress levels.

specimens gradually over a very short time span of the order of 3 s.

5. Results and discussion Experimental results for both creep and recovery viscoelastic strain response of the material tested vs. time, are

given in Figs. 2 and 3, respectively. From these figures it is obvious that the CF/Epoxy composite system tested exhibited strong viscoelastic behaviour. As a result of the 908 fibres orientation, the composite behaviour is matrix dominated. The isochronous curves presented in Fig. 4 denote a nonlinear viscoelastic strain response for values of the applied stress higher than 30% of the static tensile rupture

Fig. 3. Creep-recovery response of the 908 degrees carbon–epoxy composite for different applied tensile stress levels.

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Fig. 4. Isochronous curves for the 908 CF/Epoxy composite at different time periods.

stress. The same result can also be extracted from the compliance variation at different applied stress levels shown in Fig. 5. Following the steps described earlier, each of the nonlinear parameters was separately estimated. A graphical presentation of the g0 variation vs. applied stress is given in Fig. 6. The decreasing values of g0 with increasing stress levels, denote that a material hardening occurs because of the instantaneous application of the load. The values of D1 0, D1 c and 1 vp were determined from the

experimental values of the creep-recovery test. It is worth noting that using the earlier analytical approach, the compliance function is not necessary to be known. As the applied stress level increases, an increase in g1 values is observed and this is shown in Fig. 7. Using the Levenberg–Marquardt least squares method, the recovery test data for each recovery experiment were fitted to Eq. (9) in order to obtain an appropriate value for as . The experimental and fitted curves of the recovery response are shown in Fig. 8. In Fig. 9 the thus determined

Fig. 5. Variation of the creep compliance at different applied stress levels for the 908 CF/Epoxy composite.

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Fig. 6. Variation of the nonlinear parameter g0, as a function of the applied stress s 0.

values of as are plotted as a function of the applied stress levels. An expected reduction in the values of as with the increase of the applied stress level was observed. The value of the exponent n is necessary to be known for the estimation of each value of as . Hiel et al. [11,12] have shown that the numerical evaluation of n through the fitting of the creep data to the formulation describing the creep part, leads to unstable values of the exponent n. Besides, the exponent n was found to be dependent on the duration of the creep test. The problem can be easily solved using the

formulation of Eq. (11) which describes the recovery part of a given creep test at a low stress level s 0. Fitting the recovery data to Eq. (11) the value of the exponent n can be determined and this value is independent of the test duration as can be seen in Fig. 10. The exponent value was found to be n ˆ 0.0911. The values of the nonlinear parameter g2 were estimated for each stress level using the proposed analytical model. The use of Findley’s power law allows the simplification of the analytical formulation of Eq. (23), so that only values

Fig. 7. Variation of the nonlinear parameter g1, as a function of the applied stress s 0.

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Fig. 8. The experimental and fitted recovery strain values for different applied stress loading s 0 ˆ 40%, 55%, 70% s u.

associating to strain response are involved. Of course, the nexponent and the value of as must be known, and this was already performed at the previous step of the analysis. Thus, the analytical formulation can easily be applied in practice while the viscoplastic response is also taken into account. Limiting values obtained from each creep-recovery experiment are only needed for the estimation of the nonlinear parameter g1, at any stress level. An increase in the values of g2 was observed with increasing applied stress. A graphical presentation of g2 as a function of s 0 is given in Fig. 11.

6. Conclusions The nonlinear viscoelastic behaviour of a 908 carbon/ epoxy composite was studied. Simple step creep-recovery experiments at different applied stress levels were executed and the characteristic nonlinear viscoelastic parameters were evaluated by means of a newly developed methodology. The present methodology is based on the well-known Schapery’s nonlinear viscoelastic model and assumes that the material time dependent compliance follows a power

Fig. 9. Variation of the shift factor as , as a function of the applied stress s 0.

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Fig. 10. Experimental and fitted recovery strain values, resulted from a creep test with applied stress loading s 0 ˆ 30% s u at different recovery periods.

Fig. 11. Variation of the nonlinear parameter g2, as a function of the applied stress s 0.

law. The viscoplastic response of the material is also taken into account. Thus, the multiple fitting procedure treatment of the creep and recovery data for the determination of the nonlinear parameters – resulting in errors owing to the mutual dependence of the parameters – is avoided.

Acknowledgements The authors are grateful to the scientific and technical support of the Composite Systems and Adhesion Research Group of the Free University of Brussels (COSARGUB). Special thanks are also given to Dr. Qin Yan and Michel Bruggeman for their help.

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[10] Zaoutsos SP, Papanicolaou GC, Cardon AH. On the nonlinear viscoelastic characterization of polymer matrix composites. Composites Science and Technology 1998;58(6):883–889. [11] Hiel CC. The nonlinear viscoelastic response of resin matrix composites, PhD thesis, 1983, Vrije Universiteit Brussel (VUB), Brussels. [12] Hiel CC, Brinson HF, Cardon AH. Nonlinear viscoelastic response of polymer matrix composites. Proceedings of the 2nd International Conference on Composite Structures, 14-16. Scotland: Paisley, 1983:271–281.