Creep analysis of polymer matrix composites using viscoplastic models

8 Creep analysis of polymer matrix composites using viscoplastic models E. Kontou, National Technical University of Athens, Greece Abstract: This chapter discusses creep behavior of polymer matrix composites in the framework of viscoplasticity. The chapter first reviews the main aspects of viscoplastic creep modeling, the state of the art, with emphasis on the tensile creep response of a glass-fiber polymer composite, considering it as a thermally activated rate process. The viscoplastic part of strain is calculated separately assuming that viscoplasticity arises mainly from the polymeric matrix. Both tensile creep and monotonic loading under the same theoretical analysis in finite deformation regime are also presented. Kinematic and constitutive descriptions are presented for a three-dimensional problem. Recent developments such as nanocomposites and future trends in creep analysis are also discussed. Key words: viscoplasticity, creep, thermal activation, composites, modeling.

8.1

Introduction

Polymeric matrix composites can be tailored to meet the design requirements, which may include low density, high strength, high stiffness, corrosion and chemical resistance. As a consequence, these materials are widely used in aerospace structures, automotive parts, marine structures, etc. Moreover, learning about their properties over a range of temperature and strain rates has been an interesting task through the last few decades. Linear viscoelastic behavior can offer a frame of describing these properties (Eyring and Halsey 1946, Ward and Handley 1993, Guedes et al. 2004). However, polymer matrix composites generally exhibit non-linear and rate-dependent behavior. This non-linearity can be treated as a kinematic or physical effect. Due to the small strain these materials usually attain, physical non-linearity is mainly taken into account, which is treated as a plastic response. Therefore, as far as fiber-reinforced composites are concerned, attempts to characterize their mechanical response in the frame of linear elasticity are not adequate. A lot of theories were developed to formulate this non-linear stress– strain relationship (Hahn and Tsai 1973, Sun et al. 1974, Dvorak and Balei-El-Din 1982, Sun and Chen 1989, Van Paepegem et al. 2006). Apart from the need to consider the general non-linear viscoelastic-viscoplastic response of polymeric composites, their creep behavior has always been a topic of particular interest, due to their use in structural applications, at both ambient and elevated temperatures. Therefore, in order to obtain a correct design of structural elements, it is important to predict the long-term mechanical behavior of these materials. 273 © Woodhead Publishing Limited, 2011

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During the last few decades, several studies have been undertaken to improve the mechanical and thermal properties, as well as flame resistance, barrier properties and electrical conductivity of polymers and conventional type polymer composites. Lately, this property upgrade has been achieved by dispersing fillers of various aspect ratios on nanoscale (Fu et al. 1992, Novak 1993, Giannelis 1996, and Zhou et al. 2007, Wilkinson et al. 2006, Chow et al. 2007, Karger-Kocsis 2008, Siengchin and Karger-Kocsis 2009). The improved dispersion of nanoparticles in a polymer matrix, as well as the interfacial interaction between nanoparticles and matrix, plays a crucial role for simultaneous enhancement of a variety of properties at low contents, not met by microcomposites. In spite of the fact that the mechanical properties of nanocomposites have been widely investigated in recent years (Zheng and Ning 2003, Ash et al. 2002, Someya and Shibata 2004), there is still a lack of information regarding their time-dependent response. Referring to creep behavior, it has been shown that polymer nanocomposites exhibit substantial creep resistance due to the restricted molecular motion by the presence of nanoparticles, but their response is still nonlinear (Pegoretti et al. 2004, Zhang et al. 2004, Shaito et al. 2006). Non-linear viscoelastic or viscoplastic behavior of polymer composites can be manifested in terms of monotonic loading, creep under constant load, stress relaxation under constant deformation, time-dependent creep rupture, and timedependent strain recovery after load removal. These effects have been extensively studied by Schapery (1997) and Drozdov (1998). In the case of a small strain regime, time-dependent response of the composite can be described by linear and non-linear viscoelasticity. However, at high strain due to high stresses, yielding or alternatively time-dependent inelastic response takes place, which needs viscoplasticity models to be introduced. According to the well-known Schapery’s formulation, the non-linear viscoelastic response of any material is controlled by four stress and temperature dependent parameters, which reflect the deviation from linear viscoelasticity. In a series of works by Zaoutsos et al. (1998) and Papanicolaou et al. (1999), a new methodology for the separate estimation of the viscoelastic parameters was developed, and a further development of this methodology has been performed. For this trend, Schapery’s non-linear viscoelastic and viscoplastic model was used to describe the time-dependent response of unidirectional glass fiber reinforced epoxy matrix composites to load in the work by Megnis and Varna (2003). The non-linear time and stress dependence of viscoplastic strain was determined experimentally in off-axis creep tests as the difference between measured creep strain and predicted linear viscoelastic response. In a recent work by Vinet and Gamby (2008), a model is developed that is applicable at the scale of a quasi-isotropic visco-elastoplastic laminate during compression creep tests. The non-linear asymmetric/anisotropic viscoplastic response of fiber composites has been analytically studied in Weeks and Sun (1998), where two rate-dependent models are introduced to model the corresponding response of a fiber-reinforced

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thermoplastic matrix. Both models were developed by using a one-parameter plastic potential function to describe the non-linear behavior. In the work of Thiruppukuzhi and Sun (2001), two viscoplasticity models were examined to characterize the rate-dependent non-linear behavior of two polymeric composite material systems, namely unidirectional fiber glass and woven E-glass. Model predictions were proved to describe well the stress–strain experimental results of off-axis composite specimens at various strain rates. In a work by Kawai et al. (2007), creep and creep-recovery tests were performed on plain coupon specimens of carbon/epoxy composite laminate at various fiber orientations. Creep simulations were attempted using the modified kinematic hardening model for homogenized anisotropic inelastic composites, in which an accelerated change in kinematic hardening over a certain range of viscoplastic strain is considered. It has been shown that the proposed model can adequately describe the off-axis creep and creep recovery behavior of the unidirectional composite system. In the frame of anisotropic viscoplasticity, a model is also introduced in the work by Saleeb et al. (2003), where typical experimental data (strain-controlled tensile) and constantstress creep tests are simulated by applying the developed algorithms. Al-Haik et al. (2001), based on an elastic-viscoplastic constitutive model proposed by Gates (1991, 1992, 1993), studied the viscoplasticity of carbon fiber/polymer composite, using load relaxation and creep data. The theoretical creep curves, generated through the material constants found from stress relaxation tests, showed a close agreement with experimental data. Therefore, the model can be a design tool for durability prediction of composites for infrastructure industry applications. From another point of view, it is interesting to mention a viscoplasticity theory based on the concept of overstress in the work of Dusunceli and Colak (2008). The small strain, isotropic, viscoplasticity theory is modified so that the influences of crystallinity content on mechanical properties of semicrystalline properties are included. Considering the semicrystalline polymers somewhat as a composite, since they consist of amorphous and crystalline phases, with different resistance to deformation, this theory can be successfully extended to isotropic polymer composites. Viscoplasticity models were also introduced by Gates and Sun (1991), and Colak (2005). A plastic model based on a one-parameter potential function, by Sun and Chen (1991), was able to describe the orthotropic rate-independent behavior of polymeric composites, while another viscoplastic model, of the same trend, was proposed by Yoon and Sun (1991), for thermoplastic matrix composites, applying the concept of effective stress and effective inelastic strain. In these models, the unidirectional fiber-reinforced composites were considered as homogeneous anisotropic continua, and a single internal variable was assumed to formulate the rate effect during deformation of off-axis specimens. Tension and stress relaxation experiments were used for the estimation of model parameters, and then the prediction of stress–strain, creep, relaxation and cyclic loading was possible at various rates and temperatures. However, as mentioned by Al-Haik

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et al. (2006), these viscoplastic models fail to predict the creep response under high thermomechanical conditions, when the material parameters are derived from stress-relaxation tests. This specific issue, of describing all aspects of inelastic deformation behavior of polymeric systems in terms of a unified model, has been an interesting topic (Krempl 1998, Hasan and Boyce 1995, Spathis and Kontou 2001). In the work by Hasan and Boyce (1995), a model was presented that can describe both types of experimental results, i.e. stress–strain at compression at various temperatures, as well as creep tests at various stress levels and temperatures. The distributed nature of the microstructural state and the thermally activated evolution of the glassy state are the main assumptions of their constitutive model. Later, to the same trend, Spathis and Kontou (2001) introduced a functional form of the rate of plastic deformation in glassy polymers, based on a mechanism of plastic deformation introduced by Oleinik et al. (1993, 1995). As was shown in this work, the deformation mechanism is the same under a constant crosshead speed experiment, as well as in a creep test, with varying stress levels over a wide range. The model was proved to be capable of predicting the nonlinear viscoelastic-viscoplastic response at both types of deformation. Information on long-term deformation and strength are normally obtained by extrapolation of short-term test data, obtained under accelerated testing conditions such as higher temperature, stress and humidity to service conditions by using a prediction model (Raghavan 1997). The accuracy of prediction depends on both the model’s accuracy and its validity in extrapolation. Most of the creep models that have been introduced to model the creep of polymer composites are mechanical analogs, hereditary integrals, Schapery’s model (1997), Findley’s approach (1949), Findley and Kholsla (1955) and thermal activation theory (Eyring and Halsey 1946, Krausz and Eyring 1975). The time-temperature superposition principle, used in most of the above models, assumes that the compliance function retains its shape with respect to the logarithmic time scale. This may not be the case for many polymeric matrices of a variety of polymer composites. Apart from this shortcoming, it may be summarized that viscoplastic creep modeling in polymer composites depends on a series of factors, such as kinematic description (small and finite strain), constitutive analysis, procedure of homogeneity in the case of anisotropy, kinematic hardening rules and a micromechanics model for linking the externally imposed strain rate with the intrinsic material response. All these aspects of analysis, with the requirement of a unified description of viscoplastic deformation (creep, stress relaxation, monotonic-cycling loading), constitute an issue of high interest.

8.2

Viscoplastic creep modeling for polymer composites

One of the most common assumptions in a material’s viscoelasticity/viscoplasticity is that strain accumulation in a creep procedure follows a thermally activated

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mechanism. Thermal activation theory assumes that the macroscopic creep rate is related to the product of the number of flow units, the average strain increment per jump over an energy barrier and the probability that each unit would undergo such a jump (Conrad 1964, Krausz and Eyring 1975). However, this approach was proven to be successful in a limited strain-rate/time scale, due to the assumption that a constant number of flow units were available for transition during creep. In the work by Raghavan and Meshii (1994), a spectrum of activation processes has been used to model a creep procedure in a wider time range. In Raghavan and Meshii (1997), the creep of unidirectional carbon fiber reinforced polymer composites, as well as the epoxy matrix, both characterized as thermo-rheologically complex, was successfully modeled in a wide range of temperature and strain. The creep procedure of these composites in a temperature range of 295–433 K and at various stress levels up to 80% of the material’s tensile strength could be described. Good qualitative agreement regarding the shape of viscoelastic functions, as well as satisfactory predictive model capability has been verified. It was found that this creep mechanism is not altered by the presence of fibers; however, creep rate and magnitude is substantially affected by increasing the resistance to cooperative chain segmental mobility of the polymer chain segments and stiffening the matrix. In the following section, 8.2.1, and referring to the small strain regime, it will be shown that thermal activation theory can be a successful tool for the description of viscoplastic response during creep for polymeric composites when it is combined with a proper mechanism for calculating the rate of plastic deformation accumulating with time. Finite deformation analysis, on the other hand, is needed for many engineering applications of polymers and polymeric composites. A lot of works have dealt with finite elastic-plastic or viscoplastic deformation of polymers, but little attention has been paid to polymeric composites, in spite of the fact that they can attain high strain values, especially at high temperatures. In Section 8.2.2, the basic principles regarding kinematic formulation and constitutive modeling in the case of finite plastic deformation are considered. The implementation of this analysis on monotonic loading and creep behavior of a fiber composite is presented.

8.2.1 Small strain framework: constitutive analysis In this section, the viscoplastic creep behavior of polymer composites is studied at various temperatures and several stress levels. It is assumed, as mentioned earlier, that strain accumulation in a creep procedure follows a thermally activated mechanism. The plastic part of the strain is separately calculated following a specific function, which is based on the distributed nature of microstructural state (Spathis and Kontou 2001, Kontou 2005). The study is made in the frame of small strain, so the additive decomposition of strain is taken into account.

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Generally, assuming uniaxial loading in the small strain regime, the total strain ε may be represented in terms of the following components:

ε = εel + εR + εp

[8.1]

where εel is the instantaneous part of strain, εp is the plastic part of strain that develops during creep procedure and εR is the recoverable strain. Following Mindel and Brown (1973), creep results can be represented by an equation of the general form: . ε = f1(T) f2 (σ/ T) f3 (ε) [8.2] where f1(T), f2(σ/T), f3(ε) are separate functions of the variables temperature T, stress σ and strain ε. According to the data of Sherby and Dorn (1956), f1(T) has the form derived for a thermally activated process: f1(T) = const × exp [–Q/kT]

[8.3]

where k is Boltzmann’s constant, Q is the activation energy, and f2(σ/T) is expressed as: f2 (σ/T) = exp[συp /3kT] exp[συs /kT]

[8.4]

with υp, υs being the pressure and shear activation volume, respectively. Ward (1971) has shown that υp is much smaller than υs, so that the measured activation volume is mostly υs. The functional form of f3(ε) is as follows: f3 (ε) = α εR

[8.5]

where α is a constant and εR is the recoverable strain. Following Eq. 8.2–8.5, creep strain rate can be written as: . ε = A exp[συp/3kT] exp[(σ – σint)υs/kT]

[8.6]

where A is a constant and σint is given by:

σint = K2 εR

[8.7]

where K2 is a constant proportional to the temperature. The quantity σint has the character of an internal stress, which opposes the applied stress. Since σint increases with strain, then the effective stress σ - σint decreases with strain and therefore creep rate reaches a minimum value. In a quite similar way, in the work by Ma and Tjong (2001), the stress dependence of creep rate has been formulated as follows:

[8.8]

where A′ is a constant, σ0 is the threshold stress, denoting that the observed deformation is not driven by the applied stress σ but rather by an effective stress

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σ – σ0. G is the shear modulus, n is the stress exponent, Q is the activation energy, R the gas constant and T the temperature. Regarding plastic strain εp, it is manifested through the non-linear creep behavior exhibited by the materials. This non-linearity is accelerated when creep tests are performed at higher temperatures. The development and growth of εp was assumed to follow a mechanism, introduced by Oleinik et al. (1993, 1995), and formulated by Spathis and Kontou (2001), which will be presented below. As in the case of monotonic loading during creep, strain is accumulated around specific regions. This accumulation takes place around a large number of voids or defects, randomly distributed into the volume of the deformed material. The total evolved deformation will consequently be distributed inhomogeneously around specific regions, in a way related with the special features of each anomaly. When the distributed elastic energy, associated with strain, around each region reaches a critical value, a non-reversible transition takes place, which manages the emergence of plastic deformations. If we accept that each one of these transitions proceeds at a certain rate, then the macroscopic plastic deformations will come out with a rate proportional to the number of simultaneously appeared localized transformations. Ongoing, it is assumed that the necessary strain accumulated around the i-region randomly selected from the statistical ensample obeys a normal Gaussian distribution determined from a mean equivalent strain µ˜ , and a standard deviation ˜s . Then, the distribution density function having the strain εi as a random variable will be given by:

[8.9]

The fraction of such processes that have enough activation energy to attain a new non-reversible state is given by the probability:

[8.10]

where the lower limit (– ∞) of integration is substituted by zero, because the standard deviation is considered to be.very small. Making the further assumption that the rate of plastic deformation Γp is proportional to the fraction of plastic transformations that have achieved . a non-reversible state, and that this transition takes place with an average rate k for every plastic transformation, then we have:

[8.11]

. The value of k can be estimated, assuming that at the onset of plastic deformation, that is, at the moment where the strain ε is equal to the mean value µ˜ , the

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. rate of plastic deformation Γ py will become equal to the evolved creep strain . rate ε :

[8.12]

then:

[8.13]

The functional form of the rate of plastic deformation, presented above, was developed for the plastic deformation of polymeric materials. This concept is now extended in the case of plastic creep strain of fiber-reinforced polymer composites, denoting this way that plastic behavior exhibited by those materials arises mainly from the viscoplastic response of the polymeric matrix. The set of equations 8.1, 8.6, 8.7 and 8.13 can now be combined for the calculation of the total creep strain ε. Elastic strain εel can be obtained from the experimental data, while the plastic strain εp is calculated via a numerical . integration of Γp. In Fig. 8.1 to 8.5 the experimentally obtained tensile creep compliance of an epoxy matrix, and the corresponding off-axis epoxy-fiber glass composites, is plotted, for various stress levels at two different temperatures (Kontou 2005). The simulated values are also presented for comparison.

8.1  Tensile creep compliance of the epoxy matrix at temperatures of 333 and 353 K and at a stress level of 2 MPa. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou (2005).

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8.2  Total creep compliance of the 15° off-axis composite, at two different temperatures 333 and 353 K at a stress level of 8.6 MPa. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou (2005).

8.3  Total creep compliance of the 30° off-axis composite, at two different temperatures 333 and 353 K at a stress level of 3.68 MPa. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou (2005).

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8.4  Total creep compliance of the 60° off-axis composite, at two different temperatures 333 and 353 K at a stress level of 2 MPa. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou (2005).

8.5  Total compliance of the 90° off-axis composite, at two different temperatures 333 and 353 K at a stress level of 2.2 MPa. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou (2005).

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Comparing the creep curves of the epoxy matrix with those of the composites, it is observed that the magnitude of strain and the way it is accelerated due to the rise in temperature in the case of the epoxy is much higher. The corresponding model parameters are shown in Table 8.1. From Table 8.1 it can be observed that the pressure activation volume remains constant for all material types, while the shear activation volume appears to have a slight decrement with increasing temperature. The quite similar values for the shear activation volume for pure matrix and the 90° off-axis specimen indicate that this response arises from the viscoplastic behavior of epoxy matrix. The mean value µ˜ of the probability density function expresses an upper strain limit, which is related with a saturated material state evolved during creep procedure. The creep compliance tensor of the unidirectional fiber composites, if the tensile creep of the off-axis composites is treated as a two-dimensional problem, is given by:



[8.14]

The experimental creep results obtained from specimens oriented at 90° and 15° to the loading axis were analyzed (Kontou 2005) with the creep model to give the analytical form of compliances S22(t) and Sθ(t). The compliance of 0° samples, i.e. S11 was taken to be a constant, equal to 0.0002 MPa–1, since no creep strain emerges for those samples, even at higher temperatures. Then the compliance S66(t) could be calculated after Hyer (1998) according to the following transformation equation:

[8.15]

Table 8.1  Creep model parameter values (reproduced from Kontou 2005) Sample

Temperature (K)

υp/3kT υs/kT A (s–1) (MPa–1) (MPa–1)

Pure resin Off-axis [15°] Off-axis [30°] Off-axis [60°] Off-axis [90°]

333 353 333 353 333 353 333 353 333 353

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

1.0 0.8 0.3 0.2 0.3 0.2 0.35 0.1 0.85 0.80

1.8 10–4 1.8 10–4 2 10–6 2 10–5 2 10–6 3 10–5 2 10–6 3 10–5 2 10–6 3 10–5

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K2

εel

µ˜

700 500 800 700 1000 800 1500 1000 1500 1000

0.0055 0.0065 0.001 0.002 0.001 0.0065 0.0015 0.0022 0.002 0.003

0.015 0.016 0.02 0.025 0.02 0.01 0.019 0.025 0.02 0.025

(MPa)

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8.6  Total creep compliance of the 60° off-axis composite, at 333 K. Thick lines: experimental data. Thin lines: model predicted results. Reproduced from Kontou (2005).

where ν12 is the Poisson’s ratio, which was considered to be constant with time, equal to 0.31. However, it must be mentioned that according to the fundamental theory of viscoelasticity (Christensen 1981), Poisson’s ratios are described as time-dependent functions. Especially during polymer matrix processing, where thermal expansion takes place, the assumption of time-independent Poisson’s ratios is unacceptable, as it is pointed out by Hilton (1998, 2001). Such an assumption may lead to a non-correct characterization of real viscoelastic materials, regarding the true behavior of anisotropic moduli or compliances and relaxation/or creep functions. Only if anisotropic or isotropic viscoelastic moduli and relaxation or creep functions are characterized by identical time functions, the corresponding Poisson’s ratios must be time-independent. On the other hand, experimental measurements of Poisson’s ratios of fiber-reinforced composites, under structural conditions, have indicated that for these materials Poisson’s ratios do not vary significantly with loading frequency, as it is reported by Mead and Ioannides (1991) and Melo and Radford (2003). Comparison between experimental results and simulated ones for laminates, based on time-independent Poisson’s ratios, has shown that this assumption does not noticeably affect the predictions. The model validity can be tested through the calculation of the creep compliance of an off-axis specimen with fibers oriented at an arbitrary angle θ to the loading axis, applying equations 8.14 and 8.15. The results obtained by this equation (Kontou 2005) are presented in Fig. 8.6 for 60° off-axis at 333 K, in comparison with the corresponding experimental data. A satisfactory agreement is observed for this off-axis sample. The same procedure has been followed to check the

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8.7  Total creep compliance of the 30° off-axis composite, at 353 K. Thick lines: experimental data. Thin lines: model predicted results. Reproduced from Kontou (2005).

model validity at 353 K. The corresponding results are plotted in Fig. 8.7 for a 30° off-axis specimen. In this case, a deviation of the order of 23% is obtained, while the shape of the calculated creep compliance is retained parallel to the experimental time scale examined.

8.2.2 Finite strain viscoplasticity The inelastic material behavior under finite strain and rotation has been the subject of many works, including Onat (1982), Loret (1983), Dafalias (1984, 1985), and Anand (1985). The kinematic description of the deformed body is based on the deformation gradient F = ∇xX, where X is the reference position of a material point, and x the current position. The multiplicative decomposition of the deformation gradient tensor F into elastic and plastic components was first proposed by Lee (1969), who introduced the concept of a relaxed intermediate configuration, represented by the plastic deformation gradient. Later, the physical significance of this intermediate configuration was debated, as this is not uniquely determined, since an arbitrary rigid rotation can be superimposed on it and leave it unstressed. The basic principles of plasticity theory for finite deformation are systematically presented by Khan and Huang (1995). Hill and Rice (1972) and Asaro (1983) formulated the case of single crystal plasticity, while Loret (1983) and Asaro (1983) have extended these ideas for the single crystal to polycrystalline materials, utilizing Mandel’s approach (1971) that introduced a triad of director vectors to monitor material orientation. The difficulty

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of assigning these vectors is overcome in the case of a single crystal, whereas the vectors are naturally defined by the lattice. Haupt and Kersten (2003) and Tsakmakis (2004) attempted to represent spatial orientation of the crystal lattice by the same internal variable, which has been introduced into the strain energy function. Through this analysis, it was possible to keep the formulation compatible with the postulate of full invariance. Anisotropic kinematic hardening, on the other hand, for yield condition of slip systems has been described, among others, by Han et al. (2004). Moreover, in a recent work by Saleeb and Arnold (2004), a specific form for the polymeric material hardening function is evaluated, having a saturation value (or limit state) associated with it. This viscoplastic formulation accounts for both non-linear kinematic hardening and static recovery mechanisms. Considering the analogies in crystal anisotropy, plastic deformation of anisotropic polymer composites can be treated in terms of crystal plasticity, remaining in the frame of finite strain viscoplasticity. The basic assumption in the constitutive formulation of finite strain elastoplasticity is the multiplicative decomposition of the deformation gradient F into elastic and plastic parts introduced by Lee (1969): F = Fe Fp

[8.16]

The plastic deformation gradient tensor Fp maps a material point from a reference configuration to a relaxed configuration, which is obtained from the reference state by purely plastic deformation and rotation. Subsequently, the elastic deformation gradient tensor Fe maps the material point from the relaxed to the current configuration, which is obtained from the relaxed by purely elastic deformation and rotation. The relaxed configuration is arbitrarily defined, since an arbitrary rigid rotation can be superimposed on it and leave it unstressed. The velocity gradient tensor L in the current configuration is defined by: L = F˙ F– 1 = D + W

[8.17]

where D, W are the tensors of the rate of deformation and material spin, respectively. To overcome the problem of arbitrariness of the intermediate configuration, Mandel (1971) proposed a triad of orthonormal vectors that represent the material substructure and follow different kinematics than the continuum. The kinematic quantities Fe and Fp are then defined in terms of the corotational rates as follows:

[8.18]

where ω is the spin of the substructure, related with the rotation of the three vectors. Following this consideration, and taking into account Eq. 8.18, the kinematics in the current configuration are given by:

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Creep analysis of polymer matrix composites D = De + Dp

287 [8.19]

where De is the elastic part of the rate deformation tensor, and Dp the plastic part of it. All these quantities are now invariant upon superposed rigid body rotation. Regarding spin tensor, Dafalias (1985) showed that the total spin of the continuum, or material spin W is expressed by: W = ω + Wp

[8.20]

where Wp is the material plastic spin tensor and ω is the substructural spin. Through this relation, which is analogous to Eq. 8.19 obtained from the additive decomposition of the deformation rate tensor, the arbitrariness of Wp can be overcome with a proper value of ω. Therefore, Dafalias (1998) introduced different ω’s associated with each internal variable. The quantities Wp and Dp of Eq. 8.19 and 8.20 will be further constitutively described. Our analysis will be applied on the tensile stress–strain behavior of off-axis composite materials, as well as on the tensile creep response, assuming that it is a plane-stress problem. Given the material anisotropy, a material axis slip system can be identified by a triad of vectors s, m, n where s is a unit vector along the fiber direction, and m is a vector perpendicular to s and n = sxm. Vector s follows the materials axis, forming an angle θ with the direction x of tension (see Fig 8.8). The plastic rate of deformation tensor Dp can then be defined by the dyadics in respect to vectors s, m, n as follows: Assuming that there is a simultaneous two-dimensional shearing deformation along the fiber axis 1, and a longitudinal deformation normal to this axis, Dp can be given additively by a combination of these two deformation procedures:

[8.21]

p with D33 being the out-of-plane plastic strain rate component of tensor Dp. p It has been assumed that D11 is equal to zero, taking into account that there is no plasticity along the fiber direction, as has also been mentioned by Sun and . . Chen (1989). The multiplicative factors γ 1p, γ 2p will be defined below, and the p p isovolume condition will be valid by taking D33 = –D22 . Accordingly, the plastic spin tensor can be obtained by the following expression:



[8.22]

where  is the plastic spin coefficient, introduced by Dafalias (1998). . . The rates of plastic shear deformation γ 1p along the fiber direction and γ 2p normal to it, are assumed to be given by the expressions:

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8.8  Schematic presentation of material axes, the direction of tension, and the relative position of unit vectors s and m. Reproduced from Kontou and Spathis (2006).

[8.23]

where c is a constant, τ is the resolved shear stress equal to (σxx/2) sin 2θ, with σxx the applied tensile stress and h is a hardening modulus, which is a material parameter. Modulus h represents the internal structure of the material and may . evolve with strain hardening. In our case it was treated as a constant. A(r) is a function that is taken to be dependent on strain rate, and σ22 is the normal stress along the axis 2. To formulate the strain rate effect, which is strongly exhibited by the polymeric . fiber composites, function A(r) will be expressed in terms of a scaling rule, valid in viscoelasticity and introduced by Matsuoka (1992). Following this rule, a . stress–strain curve at a strain rate r can be obtained from a corresponding curve at

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. a rate r0 by multiplying the stress and strain with the scaling factors

289 and

correspondingly. Therefore, function A can be scaled as follows:

[8.24]

. where k, n are constants and r0 is a reference strain rate. To this trend, the anisotropic hardening and rate-sensitive behaviors of fiber-reinforced composites has been formulated in the work by Chung and Ryou (2009a, b) by a power law type, isotropic hardening law as:

[8.25]

where σ, ε are the tensile yield stress and yield strain (effective) respectively, and the other quantities are material parameters. In the same approach, the anisotropic kinematic hardening has been formulated on a Chaboche (1986) type back stress α evolution rule:

[8.26]

where 1, 2 are the fourth-order tensors containing parameters to be experimentally determined and dε is the effective strain increment. In finite deformation, the elastic deformation is generally assumed to be smaller than plastic one. The constitutive law for Dp is the flow rule at finite deformation, while the constitutive law for De is written in the frame of hypoelasticity theory. The general form of the objective stress-rate and strain-rate relation, according to Rivlin (1955) and Truesdell (1955), is given by:

[8.27]

where  is the Gauchy stress tensor, De is the elastic part of the deformation rate o tensor D, and  is the objective rate of stress tensor. The type of objective rate that is selected may lead to different results, and remains still a problem, as is also mentioned by Khan and Huang (1995). In the following analysis, the objective rate introduced by Dafalias (1985) will be applied. Therefore, the objective rate of the Gauchy stress tensor will be given by:

[8.28]

where the substructural spin is the difference between the material plastic spin and the total spin of the continuum (Eq. 8.20). Hereafter, a constitutive equation for Wp, which appears in the objective rate, is required.

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A typical form of constitutive equations of hypoelasticity is:

[8.29]

and consequently given by:

[8.30]

where C is a fourth-order tensor, which in the case of an orthotropic material and two-dimensional problem is given by:



[8.31]

– The matrix coefficients Qij are the transformed reduced stiffnesses to x-y coordinate system, with x being the axis of tension, and are defined by Hyer (1998):



[8.32]

where the matrix coefficients Qij can be written in terms of the engineering constants as:



[8.33]

Since, in the uniaxial tension experiment, the specimen is restrained by the grips of the loading machine, the total rotation of the continuum is zero. Therefore, the total material spin W is taken to be zero, assuming that for the off-axis specimens

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examined, the end-constraint effect becomes less significant. Then, according to Eq. 8.20, we obtain: p

ω  = - W

[8.34]

Taking also into account that unit vector s remains always along axis 1, it obeys the evolution law:

[8.35]

Then, given that s = (cosθ, sinθ) in respect to the x-y coordinates, we have:

[8.36]

The constitutive laws for Dp, Wp in combination with the hardening rules and the above-mentioned constitutive equations, after applying the transformation law, lead to the calculation of stress–strain in the x-y axis loading system. For similar approaches, a kinematic description, developed by Rubin (1994a, b) is presented in the work by Spathis and Kontou (2004), while the one based on the multiplicative decomposition by Lee (1969) is used in the work by Kontou and Kallimanis (2006). The analysis presented above was proven to be capable of describing a quite broad region of strain and strain rates, starting from low to moderate strain rates up to rates three orders of magnitude higher, in a variety of fiber polymer composites, as is shown in Fig. 8.9 to 8.11 (Kontou and Spathis 2006). The model capability of simulating experimental data at high rates (400–700 s–1) by Tsai and

8.9  Stress–strain curves for [15°] at three strain rates, 10–1, 10–3, 10–5 s–1. Points: experimental data after Weeks and Sun (1998). Lines: calculated results. Reproduced from Kontou and Spathis (2006).

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8.10  Stress–strain curves for [30°] at three strain rates, 10–1, 10–3, 10–5 s–1. Points: experimental data after Weeks and Sun (1998). Lines: calculated results. Reproduced from Kontou and Spathis (2006).

8.11  Stress–strain curves for [45°] at three strain rates, 10–1, 10–3, 10–5 s–1. Points: experimental data after Weeks and Sun (1998). Lines: calculated results. Reproduced from Kontou and Spathis (2006).

Sun (2002), is depicted in Fig. 8.12 and 8.13. The plastic spin coefficient η was proven to play an important role in this non-linear finite description, even in the low strains, as it is depicted in Fig. 8.14. Moreover, the model capability has also been checked in a higher strain range, implementing this analysis on unidirectional glass-fiber-epoxy matrix composites at a high temperature of 80°C, where the

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8.12  Stress–strain curves for [15°] at 400 s–1. Points: experimental data after Tsai and Sun (2002). Line: calculated results. Reproduced from Kontou and Spathis (2006).

8.13  Stress–strain curves for [30°] at 700 s–1. Points: experimental data after Tsai and Sun (2002). Line: calculated results. Reproduced from Kontou and Spathis (2006).

contribution of Wp due to angle rotation is expected to be more essential. These results are depicted in Fig. 8.15. To further demonstrate the model capability at a wider strain range, as well as its applicability to creep data, the above equations can be applied in a creep procedure. The tensile creep experiments were performed at 80°C at a stress of © Woodhead Publishing Limited, 2011

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8.14  Stress–strain curves for [15°] at a strain rate of 10–1 s–1. Points: experimental data after Weeks and Sun (1998). Lines: calculated data with the plastic spin coefficient η = 0 (linearized model) and with η = 4. Reproduced from Kontou and Spathis (2006).

8.15  Stress–strain curves for [15°] at three strain rates and at a temperature of 80°C. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou and Spathis (2006).

8.66 MPa, for a period of 4 hours. The specific temperature was selected as that where a significant creep rate is exhibited by the materials, while it is still under the Tg of the materials examined (95°C). A typical creep curve from the off-axis specimen of 15° was obtained and plotted in Fig. 8.16 in a logarithmic time scale. Some further definitions are © Woodhead Publishing Limited, 2011



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8.16  Tensile creep compliance of [15°] off-axis composite at 80 °C and at a stress level of 8.6 MPa. Points: experimental data. Thick line: calculated results. Reproduced from Kontou and Spathis (2006).

necessary for the formulation of creep. It was assumed that there is an internal stress σint, which opposes the applied stress, expressed by:

[8.37]

where Cr is a constant proportional to temperature and εxx is the axial strain. . . Therefore, the quantities γ 1p, γ 2p are now given by:

[8.38.a]



[8.38.b]

. where r, h, c are constants. The resolved shear stress τ * and the normal stress σ *22 are accordingly given by:

[8.39.a]

[8.39.b)

Applying the whole set of the above analysis, with the proper changes for creep deformation, the axial creep strain was calculated and plotted in terms of total

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creep compliance in Fig. 8.16. The main features of the experimental creep compliance curve were captured by the model simulations.

8.3

Concluding remarks

The long-term mechanical behavior of polymer matrix composites, due to their use in structural applications has always been a topic of great importance. Viscoplastic creep modeling of polymer composites, depending on a series of factors like constitutive analysis, kinematics, micromechanical models that link the externally imposed strain rate with the material response, as well as kinematic hardening rules and anisotropy, constitutes an issue of high research interest. The non-linear viscoplastic behavior exhibited by polymer composites was mainly attributed to the polymeric matrix, and analyzed in terms of a thermally activated rate process. The creep strain was expressed by the additive form of three components: elastic, plastic and recoverable. Plastic strain was separately formulated following a functional form of the rate of plastic deformation accumulated in a creep procedure. This formalism is based on a mechanism related with the distributed nature of polymeric glassy state. The model was proved to be capable of describing the creep compliance of off-axis unidirectional glass-fiber composites at various stress levels and temperatures. A theoretical procedure was also presented, dealing with the joint study of tensile response and tensile creep behavior of polymer matrix fiber composites, in the finite deformation regime. An anisotropic model of finite viscoplasticity is described, with constitutive laws for Dp and Wp written in accordance with material anisotropy, while the constitutive law of hypoelasticity is applied in its objective form.

8.4

Future trends

The prediction of long-term creep response of polymeric composites, especially under high thermomechanical conditions, is not always possible in terms of various viscoplastic models. This inadequacy is more intense when other parameters, such as humidity, UV radiation, and physical and thermal aging are taken into account. This is mainly due to the thermorheological complexity of the polymeric matrix and its highly non-linear/viscoplastic features. In the previous paragraphs, an effort has been made to present some aspects of the viscoplastic deformation of polymeric composites, selecting the class of unidirectional fiber composites, given that the inherent anisotropy renders the problem of more interest. In the preceding analysis, both the monotonic deformation and creep response have been described with the same procedure. A possible subject of future research could incorporate other modes of deformation, such as stress-relaxation and cyclic loading. In spite of the fact that the main problems of such a description still remain, a new generation of polymeric composites, that of polymeric nanocomposites, emerges. The nanofillers

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usually result in a slightly decreased creep compliance and stable creep rate at long creep periods. On the other hand, as already has been mentioned, the non-linear/viscoelastic/ viscoplastic creep response of polymer composites arises mainly from the polymeric matrix, and under certain conditions (high stress level, elevated temperature, long-term loading) is expected to be accelerated, independently of the presence of nanofillers. Moreover, the improved creep resistance is not always systematic, due to the agglomerates of nanoparticles and poor filler/ matrix interaction. To overcome this, Zhou et al. (2007) proved that in situ grafting and crosslinking of nanosilica in the course of melt and mixing with polypropylene is an effective way to improve interfacial interaction in the nanocomposites. This procedure can further be applied in manufacturing other creep-resistant thermoplastics. In a recent work by Anthoulis and Kontou (2008), a procedure is presented which provides a better understanding of the relationship between nanoclay content/structure and final nanocomposite properties. The analysis is based on the effective particle concept analyzed by Brune and Bicerano (2002) and Sheng et al. (2004), initially presented for separate clay sheets, which was extended to involve nanosized particles or agglomerates surrounded by matrix material attached to it that possesses different properties than that of the pure matrix. This concept was satisfactorily applied on tensile tests, with the further assumption that this region around a nanoparticle undergoes viscoplastic deformation (Kontou 2007) when the imposed stress field is high enough. As a future work, this analysis could be adapted for modeling of the viscoplastic creep response of polymer nanocomposites. To this trend, some recent works deal with the creep response of polymer nanocomposites such as the work of Starkova et al. (2007), where the tensile behavior and the long-term creep of PA66 and its nanocomposites filled with TiO2 nanoparticles have been investigated. Non-linear viscoelastic models and a power law have been applied, and proved to be valid in a limited time domain, while it was shown that the smaller the nanoparticles, the higher the creep resistance. These results support the need for further exploration of the fundamental mechanism for mechanical enhancement of polymer nanocomposites. Considering all these factors, namely the quality of nanofiller dispersion, the exact features of the reinforcing mechanism, and the way polymeric matrix is affected by the nanofiller’s presence, the development of a model that can predict the viscoplastic creep performance at longer times, taking into account all kinds of structural features of nanocomposites, has been a challenging topic of research.

8.5

References

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