Experimental and numerical investigations on the time-dependent behavior of woven-ply PPS thermoplastic laminates at temperatures higher than glass transition temperature

Composites: Part A 49 (2013) 165–178

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Experimental and numerical investigations on the time-dependent behavior of woven-ply PPS thermoplastic laminates at temperatures higher than glass transition temperature W. Albouy ⇑, B. Vieille, L. Taleb Groupe de Physique des Matériaux UMR 6634 CNRS, INSA Rouen Avenue de l’Université, 76801 Saint Etienne du Rouvray, France

a r t i c l e

i n f o

Article history: Received 10 December 2012 Received in revised form 4 February 2013 Accepted 23 February 2013 Available online 14 March 2013 Keywords: A. Thermoplastic resin B. Creep C. Finite element analysis D. Mechanical testing

a b s t r a c t This study was aimed at investigating the time-dependent behavior of carbon woven-ply PPS laminates at temperatures higher than its Tg (95 °C) when matrix viscoelasticity and viscoplasticity are prominent. Creep-recovery tensile tests were carried out on [(+45, 45)]7 laminates at 120 °C in order to examine the contribution of these time-dependent effects to their mechanical response. A numerical modelling has been developed to account for these behaviors. For this purpose, a linear spectral viscoelastic model and a generalized Norton-type viscoplastic model have been implemented in the FE code Cast3m. Finally, validation tests were performed to determine the model’s ability to predict the laminates response to various loadings. The numerical simulations are in good agreement with the experimental responses even for structural testing as a creep test on a notched specimen. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Long and continuous fiber-reinforced composites are still dominated by thermosetting (TS) polymer matrices, because their hightemperature mechanical properties are higher than those of thermoplastic-based composites. In addition, TS-based composites are particularly suited for impregnation into the fibers reinforcement. Thus, TS matrix composites have been extensively used over the past 40 years for aeronautical applications. Even though they display interesting mechanical properties, they also present undeniable drawbacks, such as the need for low-temperature storage, a hard-to-control cure process, a very long curing process, and handmade draping, which causes most of the irreversible defects of the manufacturing process. One projected breakthrough in the composites industry cost equation is expected to involve the large scale manufacture of continuous fiber thermoplastic (TP) composites. Thermoplastics offer improved raw materials and processing costs, as well as improved functional performance [1]. Because the processes of material consolidation do not involve exothermic curing reactions, they can use shorter autoclave cycle times (which are ideal for large production), although the temperatures involved are generally higher than those for thermosettings [2]. The meltability of thermoplastics is also an advantage for recycling purposes. Thus, high-performance thermoplastic resins (e.g. PEEK and PPS) offer a promising alternative to TS resins [3]. Further ⇑ Corresponding author. Tel.: +33 232 959 756; fax: +33 232 959 704. E-mail address: [email protected] (W. Albouy). 1359-835X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesa.2013.02.016

growth of TP-based composites is also directly linked to the knowledge of their long-term behavior (creep and fatigue or creep-fatigue interaction). Such study is essential to consider their use in industrial application. As a result, the present work was aimed at studying and apprehending viscoelasticity and viscoplasticity in woven-ply PPS laminates at temperatures higher than their glass transition temperature (Tg). To the authors’ knowledge, most of the studies on the time-dependent behavior of TP-based laminates have dealt with responses at temperatures lower than the glass transition temperature, for which there are not a lot of experimental results [4,5], making the present work quite original. This work follows an experimental study [6] on the mechanical properties of carbon fibers reinforced TP laminates subjected to severe environmental conditions (120 °C after hygrothermal aging). From the following literature review, it appears that, for an appropriate use of TP-based laminates, there is a need for the development of comprehensive material property characterization techniques and analytical modelling methods to predict their longterm mechanical response. It is particularly relevant at high temperatures, when it approaches the material’s Tg. Here are the main objectives of the present work. Section 2 deals with the background of this study from experimental and modelling standpoints, as well as with the constitutive incremental laws proposed in [7] to account for the viscoelastic viscoplastic behavior of UD-ply TS-based laminates. In Section 3, the studied material (Woven-ply C/PPS laminates) is presented, including the experimental set-up as well as the experimental responses of C/PPS laminates subjected to different thermomechanical loadings. In Section 4, the

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incremental equations implemented into the finite element code Cast3m are given as well as the model parameters’ identification and the model validation. Thus, what knowledge may be gleaned from this work rests on the experimental investigations on the time-dependent behavior of emerging woven-ply TP-based laminates, and the applicability at T > Tg (when time-dependent behavior is exacerbated) of a viscoelastic and viscoplastic model found in the literature. 2. Background The brief literature review presented in this section focuses on the viscoelastic and viscoplastic behaviors of TP-based laminates from experimental and modelling standpoints. 2.1. Experimental standpoint A possible disadvantage of PMCs is that the physical and mechanical properties of the matrix deteriorate significantly over time due to environmental factors [8], such as elevated temperature [9–12]. Depending on the loading and temperature conditions to which PMCs are subjected, viscoelasticity or viscoplasticity manifests itself in various ways, including creep under constant load, stress relaxation under constant deformation, time-dependent strain recovery following load removal and creep rupture [13]. Also, the time-dependent behavior of TP-based composites strongly depends on the duration and the loading rate [14,15]. This dependence becomes even more critical as the temperature approaches the glass transition temperature [16–18]. It is usually observed that the behavior of fibers (carbon or glass) can be considered as time independent and as such, relatively insensitive to creep [19], making the time-dependent behavior of PMCs essentially related to the matrix. Thus, the laminates lay-up and especially the fibers’ orientation with respect to the loading direction will be of utmost importance in the time-dependent response. In addition, the reinforcement architecture (UD, weave, etc.) significantly influences the material behavior. However, most of the studies available in the literature are devoted to unidirectional TP-based laminates [9,10,12,14,15,17,20]. However, some authors performed creep tests on epoxy (TS) based composites that can be easily extended to TP-based composites. Guedes and Vaz [21] compared the creep bending behavior for UD and woven reinforcement: UD laminates display lower creep strains and seem to be more stable than woven fabrics because of the absence of tows’ overlapping areas. It is also well known that time-dependent behavior is intimately related to environmental conditions. The lack of cross-links between molecules is one of the reasons why TP matrices and composites are highly dependent on temperature [22], and exhibit a more significant viscoelastic and/or viscoplastic behavior [8,20,23]. This dependence becomes more critical as temperature approaches the glass transition temperature [10,18]. Katouzian et al. [24] studied the temperature effect on the creep behavior of PEEK (TP) and Epoxy (TS) resins in two configurations: plain or reinforced by carbon fibers according to two stacking sequences [904]s and [±454]s. They emphasized a creep behavior about twice as significant for the TP resins and composites as compared to the TS ones at 23 and 120 °C. They also analyzed the instantaneous and transient creep behavior for the studied materials. For both materials, it appeared that the instantaneous behavior is almost temperature-independent whereas the transient is highly temperature dependent especially for angle-ply laminates [±454]s. The effect of temperature was also investigated by Vieille and Taleb [25] on PPS resin and woven-ply PPS-based composites. They suggested that an exacerbated ductility coming along with a large rotation of fibers in C/PPS angle-ply laminates at a temperature

higher than its Tg. Ningyun and Evans [26] shows that at high temperature and high stress levels, C/PEEK laminates are characterized by a high creep strain and a time-dependent damage using 3-point flexure creep tests. 2.2. Modelling standpoint For relatively low loading levels, the time-dependent behavior is usually described as linear viscoelastic. When the strain exceeds 1% or 2%, this viscoelasticity becomes nonlinear in most of the polymeric materials. For higher loads, most viscoelastic models lose their accuracy. Indeed, as the load increases, the viscoplastic contribution becomes predominant. Similarly, most of the viscoelastic and viscoplastic modelling lose their accuracy in the vicinity of Tg [17,18]. 2.2.1. Viscoelastic behavior From the simplest rheological formulation to the more complex approach of Schapery’s model [19], most of the viscoelastic models were established for polymers and were later adapted and extended to composite materials. One of the first formulations was a simple empirical power law to predict linear viscoelastic behavior [17]. However, the formulation validity was restricted to low stress levels. A few micromechanical modelling approaches have been developed for predicting effective viscoelastic behaviors of particle reinforced composites. The composites usually consist of perfectly rigid spherical inclusions embedded in a linear or a nonlinear viscoelastic matrix. A comprehensive review of micromechanics methods for determining effective properties of multiphase viscoelastic composites has been proposed in [27]. Approaches based on simplified micromechanical models have been developed for predicting the nonlinear viscoelastic responses of polymers reinforced with solid micro particles. Levesque et al. [28], while using the homogenized micromodel of Mori and Tanaka [29], proposed a linearized homogenization scheme for predicting nonlinear viscoelastic responses of particulate reinforced composites. The particle was modeled as linear elastic, whereas the Schapery nonlinear viscoelastic model was applied for the matrix phase [30]. The micromechanical model developed by Muliana et al. links particle and polymeric matrix responses to the homogeneous viscoelastic behaviors of composites and simultaneously recognizes the constituent (particle and matrix) responses from the overall composite responses [31]. This model proved to be capable in predicting creep compliances at different temperatures. Original approaches based on micro/macroscopic models and the Laplace transform were developed [32,33]. Lahellec et al. considered the effective behavior of composites made of linear viscoelastic matrix, for which two classical rheological models (a Maxwell matrix and a generalized Kelvin–Voigt matrix) were considered, whereas the fibers behavior was assumed to be linear elastic. Theoretical results and predictive schemes initially developed for elastic composites can be extended to viscoelastic ones by the correspondence principle [34]. In this approach, the equations governing the local state and the effective properties of linear viscoelastic composites are, after application of the Laplace transform, completely parallel to those of linear elastic composites with complex moduli. Despite all the findings made possible by the Laplace transform and the correspondence principle, this method is costly from a computational standpoint. Phenomenological constitutive theories based on internal variables have proved to be more flexible and effective [33]. Indeed, most of the authors adopt a nonlinear viscoelastic formulation, developed by Schapery [30] and based on fundamental thermodynamical laws. This formulation was widely used to predict the timedependent behavior of composite materials [35]. Megnis and Varna presented an experimental methodology to identify nonlinear vis-

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coelastic parameters and tested it on unidirectional glass fiber/ Epoxy laminates [36]. For example, Dillard et al. [37] and Papanicolaou et al. [38] studied the time-dependent behavior of C/Epoxy laminates by using this formulation. In addition, many authors tried to improve Schapery’s model to enlarge its application field, which is why Haj-Ali and Muliana [39] presented a numerical FE formulation and validated it with creep and relaxation tests on a TP matrix (PMMA) and Pasricha et al. [40] incorporated the effect of physical aging with a scaling time coefficient according to Struik’s effective time theory [41]. Nevertheless, most of these studies are limited to UD reinforcement and off-axis loadings. Some weaknesses of the Schapery approach can be pointed out as the relative complexity to adapt a well identified uniaxial formulation to get a multiaxial one, and also to take damage into account in the model’s formulation. Another viscoelastic modelling has been developed for polymer materials, and consists in generalizing the sets of elementary rheological models directly by using a ‘‘spectrum’’ of variables, for which the parameters identification is ‘‘lighter’’ because all parameters are linked by a continuous function [42]. Known as the viscoelastic spectral approach [43], this formulation was extended and validated on PMCs by Maire [44]. It relies on a spectral distribution of elementary time-dependent mechanisms, and on a relaxation time lay-out. The main advantages of this model are the reduced number of parameters, and a more direct formulation in the case of multiaxial loadings [7]. This formulation has already been applied to TS-based composites [45]. One of the purposes of this study is to examine the applicability of this approach in the case of semi-crystalline TP-based composites at a temperature higher than Tg when time-dependent behavior is exacerbated. To a first approximation and for the sake of simplicity, a linear formulation is adopted for numerical implementation consideration even if the material viscoelasticity is known to be nonlinear. 2.2.2. Viscoplastic behavior Schapery’s viscoelastic model has been associated with simple viscoplastic formulations (Lai and Bakker [46] or Zapas and Crissman [47] formulation) in order to more accurately predict the behavior of polymeric materials [48]. However, very few authors used this approach in TP-based composites [49,50]. Over the past few decades, several phenomenological models have also been developed to describe the viscoplastic behavior of PMCs [9– 11,51], for which a comprehensive review can be found in [8]. A simple one-parameter plastic formulation was developed by Sun et al. [52] to account for the time-independent behavior of PMCs, and then extended to elastic-viscoplastic behavior by Gates et al. [51]. Based on the same principle, other numerical models included the material nonlinearity, effect of temperature and rate dependence [10,17,53–55]. As an alternative to traditional explicit constitutive modelling, several investigators recently suggested what might be called implicit constitutive modelling based on artificial neural networks (ANNs) which can directly map the behavior of a viscoplastic material [56]. It has been used to account for the nonlinear viscoelastic behavior of polymeric composites [8]. The predictions of the ANN model are found to be more accurate over a wider range of stress and temperature conditions than those of the explicit nonlinear viscoelastic model, in particular near the glass transition temperature. However, they are much more complicated to implement than conventional models. Taking into account the time and temperature dependency of C/TP laminates at temperatures higher than Tg, several formulations are based on nonlinear isotropic and kinematic hardenings [9]. A generalized Norton-type model integrating the concept of the elastic domain is also used to report the plastic strains delay [57]. This model is based on a kinematics hardening, usually used when the material

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is subjected to cyclic loading, and on a constant yield strength. The rate-dependent problem and its numerical implementation are addressed within the framework of the generalized standard materials [58]. 2.2.3. Constitutive laws of the viscoelastic viscoplastic model In this paper, tensors will be underlined in direct notation: ð:Þ represents a second order tensor and ð:Þ represents a fourth order tensor. Their juxtaposition implies the usual summation operation. A superposed dot indicates the rate, a superpose 1 the inverse and a superposed T the transpose. I and I are, respectively the second and the fourth order identity tensors. The contracted product of second order tensors A and B is such as: A : B ¼ Aij Bij ¼ T AB. To a first approximation, a small strain formulation is considered. Under this assumption, the total strain can usually be divided in three parts, whose contribution is more or less important according to the loading levels: an elastic one ee , a viscoelastic one ev e and a viscoplastic one ev p . In the ply orthotropy frame, the total strain can be commonly split into:

e ¼ ee þ ev e þ ev p

ð1Þ

A linear viscoelastic spectral model introduced by Maire [44] was chosen to simulate the viscoelastic behavior of TP-based laminates at temperatures higher than Tg when time-dependent behavior is exacerbated. To a first approximation and for simplicity reason, a linear formulation is adopted for implementation considerations even though the material viscoelasticity is known to be nonlinear. The viscoelastic spectral model lays on the decomposition of the ve viscoelastic strain rate e_ in elementary kinetics ni associated with relaxation time spectrum such as:

e_ v e ¼

nb X

ð2Þ

i¼1

ni ði 2 NÞ is a set of second order tensors corresponding to viscoelastic flow elementary mechanisms, and can be seen at a set of internal variables. The viscoelastic formulation assumes a Gaussian distribution of the relaxation mechanisms weights li (see Fig. 1). The total number of mechanisms nb is set to 30 to have an overall view of the mechanisms [44]. One of the advantages of the spectral formulation is the limited number of parameters to describe the spectrum: n0 its standard deviation and nc its average. From a physical point of view, n0 gives an enhanced effect on late mechanisms and increase of nc tends to homogenize the weights. The relaxation time si and the relaxation mechanisms weight li are obtained from a Gaussian distribution of the relaxation mechanisms (see Fig. 1) such as:

Fig. 1. Gaussian spectral layout of the time-dependent relaxation mechanisms.

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si ¼ expðni Þ

ð3Þ

li

ð4Þ

 2 ! 1 ni  nc ¼ pffiffiffiffi  exp  n0 n0 p

time-independent plasticity expression is recovered. With respect to the viscoplastic yield function fv p ðr; XÞ, those laws can be written as:

e_ v p ¼ k_ v p 2n0 : nb1

with ni = nc  n0+(i1)D: ith relaxation mechanism, D ¼ Time interval between two relaxation times. The distribution of the relaxation mechanisms’ weight is norPnb malized such as: i¼1 li ¼ 1. Once the mechanisms are identified on the Gaussian distribution, they must comply with the following differential equation deriving from a thermodynamical potential [7]:

1 n_ i ¼ ðli Sve r  ni Þ

ð5Þ

si

where r is the Cauchy stress tensor and Sve is the viscoelastic compliances tensor. In the case of a woven ply, it can be defined by:

S

ve

 0  0   0 ¼  0  0  0

0 0 0

0

0 0 0

0

0 0 0 0 0 b44 =G12

0 0

0 0 0

b44 =G12

0 0 0

0

 0  0   0  0   0  0

ð6Þ

ð7Þ

The contracted product of tensors is such as:

ðr  XÞ : M : ðr  XÞ ¼ ðrj  X j ÞM ij ðri  X i Þ ¼ T ðr  XÞMðr  XÞ where M is a fourth order tensor describing the anisotropy of viscoplastic flow in shear loading [7]:

 0  0   0 M ¼  0  0  0

 0 0  0 0 0 0 0   0 0 0 0 0  0 0 1 0 0   0 0 0 0 0  0 0 0 0 1 0 0 0

ð8Þ

The constitutive laws are derived from a thermodynamical potential such as:

e_ v p ¼ Khfv p iN

@fv p @r

and a_ ¼ Khfv p iN

@fv p @X

ðr  XÞ Mðr  XÞ ðr  XÞ

and a_ ¼ k_ v p

and a_ ¼ k_ v p

Mðr  XÞ ðr  XÞ Mðr  XÞ ðr  XÞ

ð9Þ

with N a parameter representing the material rate sensitivity and K, a parameter seen as a penalty coefficient. When K ? 1, the classical

vp

ð10Þ

vp

ð11Þ

¼ e_

¼ e_

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vp vp is the Lagrange viscoplastic multiplier where k_ v p ¼ T e_ M1 e_ homogeneous to a strain rate. The interpretation of the rate-dependent phenomenon as a penalty regularization of the rate-independent one is applied [59]. Thus, the concept of ‘‘dynamic’’ yield function allows us to use the postulate of maximum dissipation in a straightforward manner, and to derive the ‘‘static’’ yield function in a consistent manner [7,60]: 1 _ _ N fvdyn p ðr; X; kv p Þ ¼ fv p ðr; XÞ  ðkv p =KÞ

where b44 is a material time-dependent parameter and G12 is the shear modulus. Such a tensor is based on the following assumptions: the individual lamina displays an elastic behavior in the fibers directions (i.e. direction 1 and 2 for a woven-ply lamina), and the stress state is plane (case of a thin laminated plate). The three parameters n0, nc, b44 are identified from a purely viscoelastic creep test. In order to develop a satisfactory modelling at high stress levels, the viscoelastic model is complemented with a viscoplastic one. The viscoplastic behavior is activated only when the stress exceeds the yield strength sy(T) (temperature dependent), which is assumed to be the same for plasticity and viscoplasticity in PMCs. In addition, the elastic and viscoplastic behaviors are supposed to be independent of each other for the considered loading rates, and the restoring phenomena is negligible. In the case of a linear kinematic hardening, the thermodynamic force X associated with the kinematic hardening variable a is defined by X ¼ da. The elastic domain is defined by the following viscoplastic yield function:

fv p ðr; XÞ ¼ ðr; XÞ  sy ðTÞ with ðr  XÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðr  XÞ : M : ðr  XÞ

e_ v p ¼ k_ v p

Mðr  XÞ

ð12Þ

This approach is used to derive the generalized Norton-type viscoplastic model, which contains inviscid plasticity as a special case when the total strain rate becomes infinitely small. Thus, the solution of the viscoplastic constitutive problem can then be issued by following the same reasoning used in time-independent models [7]. 3. Experimental investigations 3.1. Materials and specimens The studied composite material is a carbon fabric reinforced PPS prepreg laminate supplied by the company Aircelle. Each ply consists of a balanced fabric with 0 and 90° oriented fibers. The reinforcement is a 5-harness satin weave of T300 3 K carbon fibers supplied by the company Soficar, and the matrix is a high performance TP (PPS) supplied by the company Ticona, referenced as Fortron 0214. The matrix has a glass transition temperature of 95 °C and a crystallinity of 30% (DMC and DMTA tests were performed at the Crismat/CNRT laboratory in Caen). The volume fraction of carbon fibers is 50%. The consolidated plates were obtained from hot pressed prepreg plates according to a [(0, 90)]7 stacking sequence. The test specimens were cut from 600  600 mm2 plates at a minimum distance of 30 mm from the plate edges by waterjet (see Fig. 2b). For tensile tests, straight specimens with end tabs (see Fig. 2a) have been used in accordance with the EN6031 and EN6035 standards [61,62]. For creep tests, dog-bone specimens (see Fig. 2a) in accordance with the ASTM-D638 standard [63] are more suitable than straight specimens with end tabs to investigate the long-term behavior as reported in [64]. Even though the [(+45, 45)]7 angle-ply lay-up is not suitable for use in aircraft structures, it allows the authors to specifically investigate the contribution of the PPS matrix to the high-temperature response of PPS-based laminates. Such a sequence is also well suited for studying the time-dependent behavior of an angle-ply lamina at temperature higher than Tg, in order to evaluate its contribution to the time-dependent response of a Quasi-isotropic laminates when subjected to other types of loadings. This sequence complies with the Aircelle company specifications and is commonly used in aircraft nacelles’ applications in the case of thermosetting matrixbased composites. The dimensions (thickness t and width w) of each specimen were averaged from 5 measurements (see dimensions in Fig. 2a). 3.2. Experimental set-up and methods All the tests were performed using a 100 kN capacity load cell of a MTS 810 servo-hydraulic testing machine at room moisture

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Fig. 2. Dimension of the used specimens for tensile and creep tests (a), cutting pattern of off-axis specimens in a plate (b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(50%). The temperature control system includes an oven and a temperature controller, but no humidity controller. 3.2.1. Tensile tests The specimens were subjected to tensile loadings at a displacement rate V and at 120 °C (see Table 1). V denotes constant crosshead speed applied to the specimen during displacementcontrolled tests. Four specimens were tested for each test, and the average values are reported in Table 2 with their corresponding standard deviations. For tensile tests, strains were evaluated with strain gages. In accordance with the above standards, the longitudinal and shear modulus were calculated from the following definitions:

Ex ¼

DF S:De

and G12 ¼

DF 2S:Dc12

ð13Þ

where DF is the difference in the applied tensile loads at (eX)2 = 0.25% and at (eX)1 = 0.05%; S = wt the specimen’s cross section; Dc12 = DeX  Dec represents the shearing distortion and DeX = (eX)2  (eX)1 (difference in the longitudinal strains) and Dec (difference in the transversal strain) are calculated from the values given by strain gages localized along X and Y axis (see Fig. 2). As it is highlighted in the ASTM-D638 standard, the stress– strain relations of angle-ply polymer-based laminates at T > Tg do not conform to Hooke’s law throughout the elastic range but

Table 1 Standard, dimension and test conditions. Test

Standard

Test conditions

Tensile EN 6035 V = 0.5 mm/min [(0, 90)]7 stacking sequence Off-axis tensile EN 6031 V = 1 mm/min [(+45, 45)]7 stacking sequence Creep-recovery ASTM D638 [(+45, 45)]7 stacking sequence

Table 2 In-plane shear properties of C/PPS at 120 °C.

Value Standard deviation

ru12 ðMPaÞ

G12 (GPa)

eu12 ð%Þ

159 ±3.92

1.35 ±0.15

27.34 ±0.27

deviate therefrom even at stresses less than the elastic limit. For such materials, the slope of the tangent to the stress–strain curve at a low stress is usually taken as the modulus of elasticity. Since the existence of a true proportional limit in plastics is debatable, the propriety of applying the term ‘‘modulus of elasticity’’ to describe the stiffness or rigidity of such materials has been seriously questioned. The exact stress–strain characteristics of these materials are strongly dependent on factors such as the rate of stressing, temperature, and previous specimen history. However, such a value is useful if its arbitrary nature and dependence on time or temperature are realized. In the present case, when evaluating the shear modulus G12 from stress–strain curves, it is not possible to dissociate the elastic part and the viscoelastic part from our tests. What has been done here is a rough estimate of the elastic shear modulus, for a lack of better means to do it. In addition, for numerical modelling purposes, it is required to determine a threshold from which the viscoplastic behavior is activated. As illustrated in ASTM-D638 standard, an offset yield strength can be defined. The measurement of the offset yield strength is useful for materials whose stress–strain curve in the yield range is of gradual curvature. The specified value of the offset must be stated as a percent of the original gage length in conjunction with the strength value, 0.1% for instance. 3.2.2. Creep tests The creep-recovery tests performed at 120 °C consist of a 24 h creep followed by a 48 h recovery. The loading-rate was set to 50 MPa/s and the unloading time at 100 s to avoid compression over-shoots and buckling problems. It’s worth noticing that the creep tests’ temperature is about 25 °C higher than the material’s Tg. Such a temperature was chosen because advanced aeronautics structures, and particularly nacelles require high-performance fiber-reinforced polymer matrix composites, which can be used at temperatures up to about 120 °C. During creep tests, longitudinal strains were measured with an extensometer (gage length l0 = 25 mm). 3.3. Results and discussion The tensile mechanical behavior at 120 °C of both plain PPS resin and woven-ply C/PPS [(+45, 45)]7 can be observed in Fig. 3. It

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Fig. 3. Tensile mechanical behavior: [(+45, 45)]7 woven-ply C/PPS laminated vs PPS resin at 120 °C. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

clearly appears that, C/PPS laminates subjected to monotonic offaxis tensile loadings display a highly strong matrix-dominated response resulting in an elastic–ductile behavior. The highly ductile behavior of C/PPS composites at 120 °C promotes the matrix plasticization as well as the fibers rotation [25]. Ultimately, this rotation comes along with a necking phenomenon that induces a stress relaxation in the specimen (for displacement-controlled tests). The stress drop can be clearly observed in responses of angle-ply laminates subjected to monotonic tensile loadings at 120 °C (see Fig. 3). The time-dependent response of PMCs is related to the inherent time-dependent behavior of the polymeric phase and the temporal behavior of carbon fibers due to fiber–matrix debonding, interlayer delamination. Because all the later damage mechanisms are also time-dependent and occur co-jointly with polymeric creep, it is extremely difficult to separate out the individual contributions of the composite constituents [8]. Thus, in order to primarily observe the viscoelastic and viscoplastic behavior, creep tests were conducted at stresses lower than an approximate damage threshold, which was roughly determined from load–unload tests performed at 120 °C on angle-ply laminates at a load rate 0.3 MPa/s. It is important to underline that the load rate remains the same for each load-unload, hence justifying that the decrease in the stiffness of the laminates may be ascribed only to possible damage, and not to time-dependent effects. Load–unload tests consist in performing a gradual tensile loading, and in evaluating the loss in laminates’ stiffness from which the approximate damage threshold can be obtained. The test was carried out by increasing, the stress level by steps of 5% of ultimate tensile stress (160 MPa) until final failure (see Fig. 4). The highly ductile behavior of CPPS at T > Tg makes the laminates’ stiffness difficult to evaluate. The most accurate method to measure stiffness consists in identifying a linear zone for each loop with a calculation of a correlation parameter between the experimental points [65]. Each zone was considered linear for a correlation parameter higher than 0.995. The longitudinal modulus was calculated from the definition given in EN 6035 standard [62]:

E X ji ¼

DF i S  Dei

For each loop on the stress–strain diagram, the damage variable di can be calculated from the measurement of the longitudinal stiffness EX|i after each unload (see Fig. 4):

 di ¼ 1 

EX ji E X j1

 ð15Þ

Even though this method is a rough estimate for reasons of its simple expression compared to the complexity of damage mechanisms in composite materials, it’s acceptable to a first approximation in order to evaluate damage within the laminates during the test. Damage onset can be associated with a 5% loss in the material’s stiffness (see the dotted line in Fig. 4). Thus, the damage threshold is approximately equal to 80 MPa (50% of the ultimate tensile strength) for angle-ply laminates. Considering this threshold, eight creep/recovery tests have been conducted to investigate the undamaged viscoelastic/viscoplastic behavior of angle-ply laminates. The results are presented in terms of strain vs time curves (see Fig. 5). Time-dependent irreversible effects can be dissociated from the total strain (creep strain reach before recovery) by considering the residual strains after recovery. As expected, creep test results show a highly time-dependent behavior for [(+45, 45)]7 laminates at a temperature higher than Tg. After 24 h, the maximum creep strain increases regularly with the loading level from 0.4% to about 9% for 80 MPa. This total strain usually consists of an instantaneous strain (induced by the loading) and a transient one, which is divided in a reversible part (viscoelasticity) and irreversible part (viscoplasticity). Each strain contribution is measured from Fig. 5 and reported in Table 3. Viscoelastic strain is deduced by subtracting the other contributions from the total strain. It turns out that the instantaneous effects represent about one half of the maximum creep strain for the studied range of stress levels. The other half is mostly viscoelastic under a creep stress level of 50 MPa. Higher than this, viscoplasticity becomes predominant with about 35% of the total creep strain. In addition, the response to a 10 MPa creep test is almost purely viscoelastic, confirming that the viscoplastic flow threshold is around this stress level at 120 °C.

4. Numerical modelling

ð14Þ

where DFi and Dei are the differences in the applied tensile loads and longitudinal strains in the linear zone of the ith loop, S is the specimen cross section.

4.1. Time-discretization The above constitutive laws (see Section 2.2.3) have been timediscretized using a backward Euler method and implemented into

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W. Albouy et al. / Composites: Part A 49 (2013) 165–178

Fig. 4. Load/unload tensile tests on angle-ply C/PPS laminates at 120 °C and damage vs increasing applied stress. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Creep/recovery tests on angle-ply C/PPS laminates at 120 °C. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 3 Instantaneous, viscoelastic and viscoplastic contribution in total creep strain. Stress (MPa)

10 20 30 40 50 60 70 80

Creep strain (total strain)

Instantaneous strain

Value (%)

Value (%)

% of creep strain

Value (%)

% of creep strain

Niveau (%)

% of creep strain

0.37 0.77 1.56 2.17 4.39 5.99 7.36 8.95

0.17 0.36 0.71 1.00 2.03 2.71 3.42 4.29

46 47 46 46 46 45 46 48

0.18 0.29 0.63 0.74 0.94 1.42 1.30 1.56

49 38 40 34 22 24 18 17

0.02 0.12 0.22 0.43 1.42 1.86 2.64 3.10

5 16 14 20 32 31 36 35

a FE code (Cast3m). The stability analysis of this method was done by Boubakar et al. [7]. Within the framework of an incremental method associated with a Newton iterative scheme, the material state has to be calculated on a time interval [tn, tn+1] from a strain increment De, knowing the previous converging state: ðn r; n ni ; n XÞ

Viscoelastic strain

Irreversible strain

þDe ! ðnþ1 r; nþ1 ni ; nþ1 XÞ with n ð:Þ and nþ1 ð:Þ respectively the previous convergent quantities and the current ones. The state of the material at the end of time step tn+1 is calculated by a classical return-map algorithm (Table 4). It consists in an elastic prediction, which can be followed by a viscoplastic correction to comply with

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Table 4 Return-map algorithm. Calculation of the trial viscoelastic stress nþ1 r in effective stress space If fv p ðnþ1 r ; n XÞ < 0 THEN the state of the material is viscoelastic during the increment  Calculation of the elementary viscous mechanisms nþ1 ni  nþ1 r ¼ nþ1 r  nX ¼ nX ELSE Viscoplastic correction  Calculation of the viscoplastic multiplier nþ1 k_ v p forcing the condition fvdp ðnþ1 r; nþ1 X; nþ1 k_ v p Þ ¼ 0 at the end of the increment

1. Elastic–viscoelastic prediction 2. Viscoplastic yield function test

 Update of END IF

nþ1

r; nþ1 X and nþ1 ni

the yield function. nþ1 k_ v p determined by enforcing the ‘‘dynamic’’ yield condition at the end of the time step i.e. f d ðnþ1 r; nþ1 X; nþ1 k_ v p Þ ¼ 0. If the yield condition is violated, a viscovp

plastic correction is necessary. On the time interval [tn, tn+1], the time-discretization of constitutive viscoelastoplastic laws gives the following incremental laws: nþ1

r ¼ n r þ ADe  ADeve  ADevp

nþ1 X

ð16Þ

¼ n X þ dDevp

ð17Þ

The viscoelastic and viscoplastic strain increments are given by:

Dev e ¼

nb X eD  l

i

i¼1

1 þ eD

Sve ðnþ1 r Þ 

nb X

eD

i¼1

1 þ eD

ð18Þ

n ni

And

Devp ¼ Dt:nþ1 k_ v p

Mðnþ1 r  nþ1 XÞ

ð19Þ

ðnþ1 r  nþ1 XÞ

By enforcing the condition fvdp ðnþ1 r; nþ1 X; nþ1 k_ v p Þ ¼ 0, at the end of the time step, we obtain:

_

nþ1 kv p

ðnþ1 r  nþ1 XÞ ¼

!1=N

K

þ sy ðTÞ

ð20Þ

Then,

Devp ¼

Dt  nþ1 k_ v p Mðnþ1 r  nþ1 XÞ _ ðnþ1 kv p =KÞ1=N þ sy ðTÞ

ð21Þ

Combining (17) with (21):

" nþ1 X

¼Z

nX

þd

#

Dt:nþ1 k_ v p ðnþ1 k_ v p =KÞ

with Z 1 ¼ I þ d ð

_

1=N

þ sy ðTÞ

M nþ1 r

ð22Þ

Dt:nþ1 k_ v p M 1=N þsy ðTÞ

Finally, we can calculate the stress state at the end of the time step nþ1 r can be calculated from (16)

nþ1

nb X

r ¼ W n r þ ADe þ A

eD

1 þ eD i¼1

n ni þ

Dt:nþ1 k_ v p AMZ n X _ ðnþ1 kv p =KÞ1=N þ sy ðTÞ

#

ð23Þ with W1 ¼ I þ

P

 nb eD li ve þ 1þeDeD n ni þ ð i¼1 1þeD AS

Dt:nþ1 k_ v p AMZ 1=N _ þsy ðTÞ nþ1 kv p =KÞ

Knowing the expression of nþ1 r, it’s therefore possible to calculate the dynamic yield condition at tn+1: – The thermodynamic force: nþ1 X – The viscoelastic flow elementary mechanisms: þ 1þeDeD  li Sve ðnþ1 rÞ

The elastic mechanical properties of the elementary ply of C/PPS [E11, E22, m12] were obtained from tensile tests performed on [(0, 90)]7 laminates. The stiffness modulus and strength in warp and weft directions are assumed to be equal (balanced fabrics in warp and weft directions). In addition, G12 and sy were identified from tensile tests on [(+45, 45)]7 laminates at 120 °C (see Fig. 3). These properties were determined from the tests standards recalled in Table 1. The viscoelastic and viscoplastic constitutive laws are built from 3 and 2 material parameters, respectively. These parameters are identified separately from two different procedures. Firstly, the viscoelastic parameters (n0, nc and b44) were identified from a 10 MPa creep test at 120 °C (see Table 5) for which the material response is virtually purely viscoelastic. A routine was implemented in Matlab to identify these parameters through the fitting of the experimental response. This method appears to be simpler in the present case than a least-square method, because only three parameters had to be determined. The experimental and numerical responses are compared on Fig. 6. For stresses higher than 10 MPa, viscoelasticity and viscoplasticity cannot be dissociated during creep tests. Secondly, the viscoplastic parameters (K and N) were identified from monotonic tensile tests at different displacement rates (50, 5 and 0.5 mm min1) and at 120 °C. The mechanical responses are restricted to stresses lower than the damage threshold for the reasons explained in Section 3.3 (see Fig. 7). It appears that the loading rate has a strong influence on the tensile response of C/ PPS [(+45, 45)]7 laminates, confirming the time-dependent behavior of TP-based laminates. In order to determine K and N, the viscoplastic yield function is enforced for a monotonic loading ðX ¼ 0Þ, such as: 1 ðrÞ  sy  ðk_ v p =KÞN ¼ 0

nþ1 kv p =KÞ

"

4.2. Parameters identification

nþ1 ni

¼ 1þ1eD n ni

ð24Þ

In the case of an off-axis tensile loading at 45°, the above equation leads to:

log½s12  sy  ¼

1 ½logðk_ v p Þ  logðKÞ N

ð25Þ

Assuming that the anelastic strain is viscoplastic for stresses higher than the yield strength sy, the Lagrange viscoplastic multiplier k_ v p is homogeneous to an anelastic strain rate e_ an 12 such as (see Section 2.2.3):

k_ v p ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Te _ v p M 1 e_ v p ¼ e_ an 12

ð26Þ

The shear strain rate in the ply orthotropy frame (1,2 – Fig. 2a) can be calculated from longitudinal e_ X and transversal e_ Y strain rates in the specimen frame (X, Y) which are obtained from strain gages stuck along X and Y directions (Section 3.2.1):

e_ 12 ¼

e_ X  e_ Y 2

¼ e_ e12 þ e_ an 12

ð27Þ

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W. Albouy et al. / Composites: Part A 49 (2013) 165–178 Table 5 Parameters identification – mechanical properties, viscoelastic and viscoplastic parameters. Mechanical properties

Viscoelastic parameters

Viscoplastic parameters

E1 (Gpa)

E2 (Gpa)

G12 (Gpa)

v12

sy (Mpa)

nc

n0

b44

d (MPa)

K

N

56.5

56.5

1,35

0.04

10

4.05

6.9

0.6

400

8.4e12

9.5

Fig. 6. Parameters identification – creep response of angle-ply C/PPS laminates at 120 °C after a 24-h loading step at 10 MPa: experience vs numerical modelling. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. Parameters identification – Load–unload tensile test on an angle-ply C/PPS laminates at 120 °C restricted to the damage threshold: experience vs numerical modelling. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

_ an _ _e _ an _ Considering that e_ X ¼ e_ eX þ e_ an X ¼ rX =Ex þ eX and eY ¼ eY þ eY ¼ an tXY  r_ X =EX þ e_ Y , it leads to:

e_ an 12 ¼

_ an e_ an X  eY 2

¼

e_ X  e_ Y 2



r_ X 2EX

ð1 þ tXY Þ ¼ e_ 12  e_ e12

ð28Þ

From the experimental data, e_ X ; e_ Y ; EX and r_ X can be determined for each off-axis tensile test at different displacement rates, it is there_ fore possible to calculate e_ an 12 ¼ kv p . Then the curve representing log[s12  sy] vs logðk_ v p Þ can be plotted (see Fig. 8b), by determining the longitudinal stress rX for a given longitudinal anelastic strain e ean X ¼ eX  eX ¼ eX  rX =EX (see Fig. 8a). The slope at the origin represents the longitudinal stiffness EX whose value depends on the displacement rate. Finally, the viscoplastic parameters (K and N) can be obtained from a linear regression representing the function log[s12  sy] vs logðk_ v p Þ such as:

log½s12  sy  ¼

1 1 log ðk_ v p Þ þ  logðKÞ N N

ð29Þ

The material parameter d (Table 5), related to the linear kinematic hardening, can be identified from a gradual load-unload tensile test. Only the first cycles (under 80 MPa) were simulated (see Fig. 9), because the model formulation does not thus far take damage into account.

4.3. Validation Once the model parameters were identified, the ability of the proposed numerical modelling to account for various types of loading was tested. Several authors used multi-step loading (creep or relaxation) to validate their time-dependent models [8,66,67].

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Fig. 8. Identification of viscoplastic model parameters from off-axis tensile tests at different strain rates: (a) Determination of the longitudinal stress rX for a given _ longitudinal anelastic strain ean X , (b) curve log[s12  sy] vs logðkv p Þ. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Parameters identification - In-plane shear tests on an angle-ply C/PPS laminates at different strain rates at 120 °C: experience vs numerical modelling. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Model validation – Multi-step creep test at 120 °C on an angle-ply C/PPS laminates. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Thus, the first validation test consisted in simulating a five-step creep test. Such a test was carried out to investigate the effect of multiple loading steps on the creep strain rate. The specimen was gradually loaded through five 24 h steps with a stress load

Fig. 11. Creep strain rates of single-step and multi-step creep tests on an angle-ply C/PPS laminates at 120 °C vs stress level applied. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

ranging from 40 to 80 MPa. The loading phase is followed by a 24 h recovery. The load rate was identical to the single creep tests. The strain vs time response of the multi-step creep test is presented in Fig. 10 as well as in the model prediction. At the same

W. Albouy et al. / Composites: Part A 49 (2013) 165–178

175

Fig. 12. Model validation – Short creep-recovery tests on an angle-ply C/PPS laminates at 120 °C – (a) Applied stress, (b) stress/strain diagrams experience vs model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 13. Meshing of one quarter of the specimen with 960 four-node shell elements. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

time, it tests the model’s ability to predict the instantaneous strain part involved after each loading and to account for the effect of multiple loadings on the strain rate. Indeed, the stress history and the time-dependent mechanisms have a strong influence on the creep kinetics (see Fig. 11) and the creep strain levels reach at each step. The comparison of numerical and experimental results shows that the modelling is in rather good agreement with the experience from qualitative and quantitative standpoints. However, Fig. 10 suggests that the model does not account for the effect of multiple loading steps on creep strain rates observed during the test. According to Petipas [68], the creep rate change is a result of the nonlinear viscoelastic behavior of the material, which is not taken into account here. A possible improvement would be to adopt a nonlinear viscoelastic spectral formulation. The second validation tests focused on the load and unload phases of creep

tests. Three different short creep tests (1000 s per step) followed by a recovery phase were carried out. The specimens were first loaded at 40 MPa, and then at 60, 80 or 100 MPa respectively, before returning to the 40 MPa stress level, and ultimately to free stress state (see Fig. 12a). The results are presented in stress–strain diagrams to emphasize time effects, and to observe how the model is able to predict the load-unload phases (Fig. 12b). From the experiment vs modelling comparison it appears that these predictions are quantitatively and qualitatively good. It is also true at the highest stress level (100 MPa) even though the applied stress is higher than the damage threshold. The numerical model has been evaluated in the case of a structural testing which consists in subjecting notched angle-ply laminates to long-term tensile loadings. For symmetry reasons, only one quarter of the specimen can be considered for simulating the response of notched angle-ply

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Fig. 14. Model validation – Creep test on a notched angle-ply C/PPS laminates at 120 °C – (a) strain vs time diagram and (b) stress vs strain diagram. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

specimens (see Fig. 13). 960 four-nodes shell elements have been used to mesh the structure, with a refined mesh near the hole. The boundary conditions are shown in Fig. 13. The response of a notched angle-ply C/PPS laminates specimen subjected to a creep-recovery test at 120 °C has been simulated. The hole diameter is set to 3 mm, making the ratio specimen width on hole diameter w/d = 5. Two successive creep-recovery tests were performed at unnotched stresses equal to 30 and 66 MPa, respectively corresponding to notched stresses equal to 40 and 80 MPa (effective section) during 24 h, such as:

runnotched ¼

F we

and

rnotched ¼

F ðw  dÞ  e

ð30Þ

The creep results can be presented in two ways: either the longitudinal strain vs time curves or the longitudinal stress vs strain curves. The comparison of the mechanical responses (see Fig. 14) ultimately shows that the viscoelastic viscoplastic model, constructed as so, is suitable for modelling the time-dependent behavior of woven-ply reinforced TP laminates at T > Tg. 5. Conclusion It is known that time-dependent behaviors (viscoelasticity and viscoplasticity) of polymer-based composites are exacerbated at temperatures higher than their glass transition temperature. The

present experimental and numerical work was therefore aimed at investigating the influence of time-dependent effects on the long-term behavior of carbon woven-ply PPS laminates at 120 °C > Tg. Firstly, and in order to limit the effects of damage on the time-dependent response of such composite materials, an approximate damage threshold (80 MPa) was roughly determined from load–unload tests, and the cre-recovery tests have been conducted under this threshold. Thus, the laminate’s response to creep tests showed that over than 50% of the total strain finds its origin in viscoelastic and viscoplastic effects. Secondly, in order to investigate more precisely the long-term behavior of PPS matrix at T > Tg, a numerical modelling consisting of a viscoelastic spectral and a generalized Norton-type viscoplastic formulation developed for TS-based composites have been adopted. The idea of the present work was to examine the applicability of this approach in the case of semi-crystalline TP-based composites at T > Tg, when most models proposed in the literature lose their predictive capacity. Thus, the proposed modelling based on a reduced number of parameters proved to be promising to adequately predict the behavior of TP-based laminated at high temperature, even in the case of a structural testing (creep-recovery on notched laminates). At last, long-term behavior of woven-ply PPS-based laminates will be further investigated through the study of the interaction of creep on subsequent fatigue behavior, still at temperatures higher than the glass transition temperature. At the same time, a nonlinear formulation of the viscoelastic model will be adapted, and

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