Calibration of a viscoplastic cohesive zone for crazing in PMMA

Engineering Fracture Mechanics 73 (2006) 2503–2522 www.elsevier.com/locate/engfracmech

Calibration of a viscoplastic cohesive zone for crazing in PMMA N. Saad-Gouider a, R. Estevez a

b

a,*

, C. Olagnon a, R. Se´gue´la

b

Groupes d’Etudes de Me´tallurgie Physique et de Physique des Mate´riaux, UMR 5510, INSA Lyon, 20 Av. Albert Einstein, 69621 Villeurbanne Cedex, France Laboratoire Structure et Proprie´te´s de l’Etat Solide, Universite´ des Sciences et Technologies de Lille, Baˆtiment C6, 59655 Villeneuve d’Ascq, France Received 1 October 2005; received in revised form 28 April 2006; accepted 5 May 2006 Available online 5 July 2006

Abstract In a numerical analysis of mode I fracture in amorphous polymers, Estevez et al. [Estevez R, Tijssens MGA, Van der Giessen E. Modelling of the competition between shear yielding and crazing in glassy polymers. J Mech Phys Solids 2000;48:2585–617] have shown that the material toughness is governed by the competition between the time scales related to shear yielding and crazing. The present study aims at calibrating the parameters involved in this description, for a commercial PMMA. An elastic–viscoplastic constitutive law featuring softening upon yielding and hardening at continued deformation is used for the bulk while crazing is described with a viscoplastic cohesive zone. The three steps of crazing with initiation for a critical stress state, thickening of the craze surfaces and breakdown of the craze fibrils for a critical opening are characterized separately. In particular, it is demonstrated that the use of a viscoplastic cohesive zone is necessary to capture the variation of the toughness with loading rate. For PMMA, the related energy release rate is shown to depend primarily on the craze critical opening and the craze thickening kinetics while craze initiation is of minor importance for the quasi-static loading conditions under consideration here.  2006 Elsevier Ltd. All rights reserved. Keywords: Crazing; Cohesive zone; PMMA; Elastic–viscoplastic; Fracture

1. Introduction Failure of amorphous polymers in the glassy state involves two mechanisms of damage and failure: shear yielding and crazing [1]. When crazing is suppressed, as in compression, shear yielding takes place in the form of a localized plastic deformation through shear bands related to the intrinsic softening upon yielding followed by a progressive strain hardening as the deformation continues. Crazing involves also some localized plasticity [2,3], albeit at a smaller scale, and is the mechanism responsible for failure. After initiation at a critical stress state, the craze thickens by the growth of fibrils of which breakdown at a critical thickness corresponds to the nucleation of a crack. In a numerical study [1] featuring a viscoplastic model for shear yielding and a *

Corresponding author. Tel.: +33 4 72 43 80 83; fax: +33 4 72 43 85 39. E-mail address: [email protected] (R. Estevez).

0013-7944/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.05.006

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viscoplastic cohesive zone for crazing, it was demonstrated that the competition between the kinetics of these two mechanisms together with the condition for the craze fibrils breakdown govern the level of the toughness, as for instance the ductile to brittle transition observed at low loading rates. A ductile response is related to the development of some plasticity in the bulk prior to crack propagation while a brittle response corresponds to the development of crazing only, the bulk remaining elastic. The present study is connected to a modelling of crazing [1,4] within a cohesive surface methodology which incorporates the three characteristic stages of crazing. Estevez et al. have shown [1] that the description adopted here is able to capture qualitatively the main features of amorphous polymers fracture as the rate dependency of the toughness. The aim of the present study is to provide some quantitative estimates of the fracture characteristics and to identify the dominant mechanism or effect responsible for the material’s toughness at a given loading condition. We present the complete experimental protocol necessary to calibrate the parameters of the description of crazing and shear yielding. The calibration is based on the analysis of fracture and uniaxial compression tests performed at various loading rates, for quasi-static loading conditions. It is shown that a viscoplastic cohesive zone model for crazing is necessary to predict the rate dependent toughness observed experimentally. Experimental measurements of the toughness at various loading rates in specimens with two different blunted crack tips appear to be in good agreement with the corresponding predictions. In particular, the model is able to capture the size effect introduced when varying the notch radius thanks to the intrinsic length scale introduced by the cohesive zone description. In order to illustrate the methodology, a commercial PMMA (Perspex) is used, which is generally thought brittle and (quasi-) linear elastic under tension. While the bulk response of PMMA can be considered linear elastic at a first approximation but with a secant Young’s modulus representing the viscoelastic effects, we show that its fracture characteristics are noticeably rate dependent. Tensors are denoted by bold-face symbols,  is the tensor product and • the scalar product. For example, with respect to a Cartesian basis ei, AB = AikBkjei  ej, A • B = AijBij and LB ¼ Lijkl Bkl ei  ej , with an implicit summation over Latin indices. The summation is not used for repeated Greek indices. A ( ) 0 identifies the deviatoric part of a second-order tensor, I is the identity second-order tensor and tr denotes the trace. 2. Bulk constitutive law 2.1. Modelling background In the absence of crazing, glassy polymers can undergo a deformation up to 100% with an intriguing constitutive law with softening upon yielding followed by hardening. In an analysis of the crack tip plasticity under a mode I loading, Van der Giessen and Lai [5] have shown that the observed softening is intrinsic to the material response and is not due to a structure or geometrical effect. Their prediction of the shape of the plastic zone and trajectories of the shear bands is in good agreement with reported observations of Ishikawa et al. [6]. The constitutive law used to model the large strain plastic behaviour is based on original ideas due to Boyce et al. [7] but with some modifications introduced later by Wu and Van der Giessen [8] for the hardening part. We present the governing equations to point out the parameters to be identified, the reader is referred to [9] for details on the computational aspects. The mechanics of fully three-dimensional large strain deformation involves the deformation gradient tensor F, which maps a material point of the reference configuration into the current configuration. The deformation gradient is multiplicatively decomposed as F = F eF p, with F p a deformation from the initial to an intermediate, ‘‘relaxed’’ or ‘‘natural’’ configuration, followed by an elastic transformation F e up to the final deforma_ 1 ¼ F_ e F e þ F e F_ p F p F e of which tion F. The velocity gradient in the current configuration is L ¼ FF symmetric and anti-symmetric part correspond to the rate of the strain and spin tensor respectively. When the elastic part Fe of the deformation gradient is small compared to the plastic one F p (i.e. F e  I), the velocity gradient results in L  Le + Lp so that the total strain rate D becomes the sum of the elastic and plastic parts as D = De + Dp. Prior to the yield stress, most amorphous polymers show a non-linear stress–strain response due to small viscoelastic effects. These are not considered explicitly but their effect on the mechanical response is accounted for by using a secant Young’s modulus instead of that derived from ultra-sonic measurements of the elastic wave velocities. Its value is estimated from uniaxial compression tests and an average value for dif-

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ferent strain rates of Esecant = ry/ey, with ry the yield stress and ey the corresponding yield strain, is used. In view of these approximations, the hypo-elastic law is used to express the bulk mechanical response as r

r ¼ Le De ¼ Le ðD  Dp Þ;

ð1Þ

r

where r is the Jaumann rate of the Cauchy stress, Le the fourth-order isotropic elastic tensor in terms of secant Young’s modulus Esecant and Poisson’s ratio m, which is in Cartesian components Lijkl ¼   secant E 2m ðdik djl þ dil djk Þ þ 12m dij dkl . 2ð1þmÞ Within the elastic–viscoplastic framework used here, the plastic strain rate Dp is 0 r Dp ¼ c_ p pffiffiffi 2s

ð2Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0  r 0 =2 with c_ p ðs; p; T Þ the equivalent shear strain rate which is temperature and stress dependent, and s ¼ r 0  corresponds to the deviatoric part of the driving or effective the equivalent shear stress. In Eq. (2), the tensor r  ¼ r  b. The back stress which is the difference between the applied Cauchy stress r and a back stress b as r stress tensor b is due to the entropic back forces generated by the deformation of the polymer chains during the plastic deformation, and will be defined later on. The equivalent plastic shear strain rate c_ p is taken according to Argon’s original idea [10] "  5=6 !# Aðs þ apÞ s 0 1 c_ p ¼ c_ 0 exp  ; ð3Þ T s0 þ ap in which A and c_ 0 are material parameters and T is the absolute temperature. The athermal shear strength s0 in Argon’s original formulation is s0 ¼ 0:077G , with G the shear modulus at high frequency and m the Poisson 1m ratio. It is worth noting here that the shear modulus G involved in (3) is not connected to the secant Young’s modulus Esecant used in the bulk constitutive law (1). The first determines the activation energy responsible for plasticity locally while the Esecant is thought to represent the influence of viscoelastic effects on the bulk mechanical response. This choice for the value of G seems more consistent than an estimate from a tensile or compression tests for which the value derived is dependent on the prescribed strain rate or stress level at which the measurement is performed. The coefficient a represents the pressure sensitivity of the polymer which results in an asymmetric yield stress in tension and compression, for instance. From a micromechanical point of view, the expression of the temperature and rate dependency of Eq. (3) is general enough to describe the variation of the yield stress of glassy polymers due to viscoplasticity and we adopt this expression even if the physical arguments underlying Argon’s view do not receive full agreement in the polymer community. Following Boyce et al. [7], intrinsic softening is accounted for with the definition of an internal variable s which varies from s0 to sss at continued plastic deformation. The internal law s_ ¼ hð1  s=sss Þ_cp governs its variation during deformation, with h a parameter controlling the rate of softening and sss the value of s in a steady state regime. The progressive hardening due to the plastic deformation and induced molecular orientation results in a back stress tensor b which can be considered as the development of internal stresses during the deformation. Its description is based on ideas borrowed from theories for rubber elasticity with the cross-links of the rubbers considered as ‘‘entanglements’’ in the case of glassy polymers [7,8]. The deformation of the resulting network is assumed to derive from the cumulated plastic stretch [8] so that the principal back stress components ba are functions of the principal plastic stretches kb as X b¼ ba ðepa  epa Þ; ba ¼ ba ðkb Þ; ð4Þ a

epa

in which are the principal directions of the plastic stretch. The estimate of the back stress b used in our description is due to Wu and Van der Giessen [8] on the basis of their analysis of the fully three-dimensional orientation distribution of molecular chains in a non-Gaussian network. Wu and Van der Giessen have shown that a fairly good estimate of the back stress tensor b can be obtained with the following combination of the three-chains and eight-chains [11] models as

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bn-ch ¼ ð1  qÞb3-ch þ qb8-ch a a a

ð5Þ

pffiffiffiffi with q ¼ 0:85 k= N ;  k ¼ maxðk1 ; k2 ; k3 Þ the maximum plastic stretch and pffiffiffiffiN the average number of segments between entanglements. The limit stretch of a molecular chain is kmax ¼ N . The expressions for the principal components of b3-ch and b8-ch contain an additional parameter: the shear modulus CR of the network taken as a a R C = nkBT, n being the entanglement density, kB the Boltzmann constant and T the temperature. 2.2. Experimental procedure In order to obtain a set of material parameters for the description of the finite strain viscoplastic response of our PMMA, experimental data of the stress–strain response for various strain rates are necessary. At room temperature, tensile experiments are unsuitable for this purpose because PMMA will fail by crazing before the macroscopic yield stress is attained. In compression, crazing is suppressed and uniaxial experiments are carried out on cylindrical specimens. In order to prevent any buckling, the diameter and the height of the specimen are 8 mm and 10 mm respectively. Low friction along the contact surface between the faces of the cylinder and the plates of the machine is ensured by using a hexagonal boron nitride powder (BN). Therefore, lateral displacement with respect to the direction of compression is allowed so that a uniform stress state prevails along the whole specimen. In order to characterize the rate dependency of the yield stress, various strain rates ranging from 1 · 105 s1 to 1 · 101 s1 are investigated. We use a linear variable displacement transducer (LVDT) to measure the axial deformation of the specimen, the lateral variation being estimated from the assumption of an isochoric deformation. All samples were compressed by a 6.5 mm amplitude (approximately 100% of strain). We note that prescribing a constant clamp displacement rate, as it is done practically, results in variable strain rate as e_ 0 eeðtÞ . This effect will be accounted for in the simulations for the identification of the bulk parameters. 2.2.1. Thermal effects Do temperature effects need to be considered or is the isothermal assumption relevant in the compression experiments? To clarify the domain in terms of loading rate for which isothermal or adiabatic conditions prevail, we consider a time scale for heat diffusion over a characteristic length to be representative of the thermal effects to be compared to a time scale related to the loading. In this way, we implicitly assume that the time scale related to heat radiation at the surface of the specimen is comparable to that of heat diffusion. Such a time scale for heat is defined by using L a relevant length scale of the problem, for instance half the diameter of the sample, so that the characteristic time for heat diffusion is theat = L2/v with v is the thermal diffusivity (v = k/qc, k the thermal conductivity, q the mass density and c the specific heat). A time scale related to the loading can be derived from the time necessary to reach 100% of plastic deformation of which only a fraction is converted to heat so that tload  1=_e with e_ the strain rate under consideration. Isothermal conditions will prevail when theat  tload, i.e. when heat diffusion is much faster than the time scale related to the loading while adiabatic conditions correspond to theat  tload, and coupled thermomechanical conditions appear when theat tload. This is the case when the strain rate e_ > 1  102 s1 for which a couple thermo-mechanical or adiabatic analysis is necessary. For strain rate slower than 1 · 102 s1, isothermal conditions hold. We will adjust the bulk parameters for these low strain rates and consider the high strain rates in a second step. 2.2.2. Identification of the bulk parameters The parameters involved in the elastic response of the material as well as the various dependencies of the yield stress are derived directly from experimental data. The stress–strain response prior to yielding is approximated by a linear secant modulus Esecant of which average value of ry/ey at the yield stress is considered, so that E secant = 2.2 GPa. The Poisson ratio m and the initial shear modulus G are derived from the measurement of the bulk and shear velocities with an ultra-sonic set-up, which results in m = 0.32 and G = 2 GPa (corresponding to a Young’s modulus of 5.3 GPa). The athermal yield stress s0 is then estimated from Argon’s expression with

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0

2507

. ln γ p

-1

-2

-3

-4

Xy -5 100

110

120

130

140

150

160

Fig. 1. Variation of the plastic strain rate with the generalised variable Xy for the five strain rates from 105 s1 to 101 s1.

s0 = 216 MPa (Section 2.1). The yield stress is sensitive to the pressure thus resulting in an asymmetric response from tension to compression and a value of a = 0.1 is used here [12]. The strain rate and temperature dependence of the initial yield stress involve the parameters c_ 0 and A of (3). p At the yield stress and prior to softening, pffiffiffi the plastic strain rate c_ equals the prescribed c_ . The following quantity are then  derived:  s ¼ sy ¼ ry = 3 and py = ry/3 which are used in conjunction with (3) rearranged as c_ p ¼ c_ 0 exp  TA X y , in which the variable Xy = (s0 + a py)[1  (sy/s0 + apy)5/6] is introduced. The plot of ln c_ p versus Xy reported in Fig. 1 provides an estimation of c_ 0 and A from a linear fit of the data. The parameters for the description of the post-yield response (softening and hardening) are derived from the best fit between the simulations of the uniaxial compression test and the experimental data. The rate of softening is controlled by h up to a steady state for which the internal variable s reaches sss. The hardening response involves the back stress characterized by the modulus CR and N. The corresponding fit is shown in Fig. 2 and the parameters are reported in Table 1. In Fig. 2a, the experimental data and corresponding simulations for isothermal conditions ð_e < 1  102 s1 Þ show that the description is well predictive for these loading rates throughout a range of deformation up to 100%. For higher loading rates, temperature effects need to be accounted for. The constitutive law (1) is r in this case r ¼ Le De  Cac T_ I, with ac the coefficient of thermal expansion, C the bulk modulus and I the second identity tensor. The estimate of the back stress is also by temperature variations and the tem affected  Ea [13,14] needs to be considered which results perature dependent entanglement density nðT Þ ¼ B  D exp  RT in CR = n(T)kT. The conservation of mass imposes the product n Æ N to remain constant [14] so that N and the related maximum stretch increase with temperature. The reader is referred to [13] for a detailed description of the governing equations of the thermo-mechanical problem. In the case of adiabatic conditions, the plastic _p¼r _ p  Dt=qc which affects c_ p ðT ; p; sÞ and 0 Dp results in the temperature variation DT ¼ D dissipation D R p _ accounts for the effective stress r 0 ¼ r0  b0 in which b(C (T), N(T)). The estimate of the plastic dissipation D p 0 the energy stored by the network b • D is not converted into heat. Therefore, only a fraction of the external work is involved in heat production. The corresponding simulations are reported in Fig. 2b where it is observed that the experimental softening is more pronounced than the predicted one. This shows a limitation of the bulk description which is not able to fully capture the thermal softening. This discrepancy due to temperature effects is unlikely to occur when crazing takes place since the presence of a craze reduces the amount of plastic deformation [15] and the related temperature increase. In a coupled thermo-mechanical analysis of the temperature effects near mode I cracks in which crazing and shear yielding are accounted for, it is shown [15] that a noticeable temperature variation is observed for a loading time smaller than 10 ms, much faster than any loading to be investigated here. Therefore, the identification for isothermal conditions is thought valid for the present study, when mode I fracture is concerned.

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σ (MPa)

150

100

Experiments

50

Simulation

ε

0 0

0.2

0.4

0.6

0.8

1

(a) 200

σ (MPa)

150

100 Experiments Simulation

50

ε

0 0

0.2

0.4

0.6

0.8

1

(b) Fig. 2. Results from compression experiments in terms of the Cauchy stress versus logarithmic strain, compared to predictions with the material constitutive law (Section 2.1), with the parameters in Table 1: (a) isothermal assumption for strain rate 1 · 105–1 · 103 s1 and (b) adiabatic assumption for strain rate 1 · 102–1 · 101 s1.

Table 1 Material parameter set for PMMA for the constitutive model described in Section 2.1 Elastic E secant 2.2 GPa

Viscoplastic m 0.32

s0 216 MPa

Softening a 0.1

A 60 K/MPa

c_ 0 2 · 10

10

1

(s )

Hardening

sss/s0

h

CR

N

0.88

900 MPa

18 MPa

3.3

3. Modelling crazing with a viscoplastic cohesive zone Motivated by the Kramer and Berger [2] description of craze thickening, Tijssens et al. [4] proposed a viscoplastic description for crazing within the framework of a cohesive zone methodology. The traction-separation law proposed in [4] comprises three parts corresponding to initiation, thickening and breakdown of a craze. The physical mechanism for craze initiation is not yet clearly identified and various criteria have been proposed depending on the assumed mechanism and length scale (see [16] for a review). Craze initiation is thought to involve some stress and temperature controlled mechanism at a critical stress state (or equivalently a critical

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strain state). The identification of the criterion and the stress state for which craze initiation appears is generally phenomenological [17,18] except a description due to Argon and Hannoosh [19] based on the idea of thermally activated nucleations of micro pores which has more physical foundations. The above mentioned criteria provide similar prediction for the initiation, at least in the first principal stress quadrant. In the context of the cohesive zone formulation, craze initiation requires (i) a positive mean stress and (ii) a critical stress rcr Ongchin’s criterion [17] can be reformun ðrm ; T Þ normal to the craze plane. For instance, the Sternstein and A0 B0 lated within the cohesive surfaces context so that [1] rcr ¼ r  þ , under plane strain conditions, with m n 2 6rm the nucleation of a craze occurring for rn ¼ rcr . The variables r and r correspond to the normal and mean n m n stresses at the plane of initiation. The coefficients A0 and B0 are material parameters which can be temperature dependent [20]. Any other definition of rcr n is acceptable, as long as the adopted criterion matches the experimental data but evidently, the mechanism underlying craze initiation needs to be further clarified. Following Kramer’s description [2,3], crazes thicken by drawing material from the bulk to the craze fibrils through a transformation which implies an intense plastic deformation within a layer of some nanometres at the bulk–fibrils interface. Based on this observation, Tijssens et al. [4] have suggested to use a viscoplastic formulation for the craze thickening as  c c A r rn

D_ cn ¼ D_ 0 exp 1 c ð6Þ T r with D_ 0 a pre-exponential term which involves a rate dependency, Ac accounts for the temperature dependence and rc is an athermal stress for fibrillation. The thickening rate D_ cn represents the relaxation of a craze subjected to a normal stress rn along its surfaces. Once crazing initiates, the thickening of the fibrils continues up to their breakdown for a critical thickness Dcr n which is material dependent. The three stages of the crazing process are combined in the traction-opening law r_ n ¼ k n ðD_ n  D_ cn Þ

ð7Þ

with D_ n the normal opening rate of the cohesive surface, D_ cn the thickening rate of the craze according to (6) and knan elastic stiffness. The traction-opening law in (7) is used for the three stages of crazing. Prior to craze initiation, D_ cn is not relevant and (7) reduces to r_ n ¼ k n D_ n in which the stiffness kn has to be ‘infinitely’ large to ensure the elastic opening to remain small and to have no significant effect on the continuity of the fields. This parameter is physically defined in [1] as k 0n ¼ rcr n =h0 with h0 about 20 nm. We are not incorporating here the details related to the variation of the stiffness kn due to the transformation of a ‘‘primitive’’ fibril at the onset of craze initiation to a ‘‘mature’’ fibril as described by Donald and Kramer [21] and incorporated in the cohesive zone description in [1]. Those details in the variations of kn have been observed of negligible influence on the present predictions of the toughness for the loading rate under consideration here. The height h0 of 20 nm is used to provide the proper order of magnitude for the stiffness of the cohesive surface. The expression (7) is used during craze thickening. At the onset of craze fibrils breakdown, the normal stress vanishes as a crack has nucleated. For an elastic material with crazing only, the energy release rate Gc is derived from the tractionopening law in (7) as Z Dcrn rn ðD_ cn ÞdDn : ð8Þ Gc ¼ 0

For a given loading rate, Gc is a measure of the energy dissipated during craze thickening. In Fig. 3, the response of the cohesive zone is shown schematically. Three parts are distinguished with a stiff response prior to craze initiation followed by a hardening-like (2a) or a softening (2b) transient regime before a steady state thickening at approximately a constant normal stress is observed, up to the craze fibril breakdown which is represented by a vanishing normal traction rn. The responses (2a) or (2b) are dependent on the condition for craze initiation and the loading rate ðD_ n Þ which determines the level for which craze thickening will take place. Recent investigations in PMMA have addressed the formulation of a cohesive zone to model failure, essentially when analyzing dynamic fracture [22,23]. Comparisons of the predicted temperature effects due to crack propagation on the crack pattern for cracks running at velocities above half CR (the Rayleigh velocity) and

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Fig. 3. Traction-opening law of the cohesive zone for crazing in glassy polymers.

crack branching during propagation with experimental observations [22] are thought to provide hints of the most appropriate formulation to adopt. Bjerke and Lambros [22] have suggested the use of a linearly decreasing traction-opening law to match their temperature record during failure. However, these authors come out with values of the energy release rate several times larger than its value under quasi-static conditions and a length of the cohesive zone about 1 mm, the peak stress at initiation being of the order of 120 MPa. Such extension of the cohesive zone is unlikely to be related to a craze which is about 30–100 lm long and some microns thick. At least, the parameters identified by Bjerke and Lambros [22] represent not only crazing but the whole thermo-mechanical process involved in the failure, as suggested by the authors to explain the dimensions they derive. More recently, Murphy and Ivankovic [23] have tried to identify the formulation of the cohesive zone which would best fit their observations of the crack pattern formed during branching in dynamic fracture. They showed that the dynamic results are well captured when using a cohesive zone with a transient softening upon initiation followed by a steady state thickening up to craze fibril breakdown. In this case, the dimensions of the cohesive zone they identified are closer to those usually reported for crazes. The cohesive surface formulation adopted here originates from physically available descriptions and is in some aspects similar to that of Murphy and Ivankovic [23], especially the description of the steady state craze thickening. However, the viscoplastic nature of our description involves possible rate dependency of the material toughness which is shown significant in the sequel. The calibration of the cohesive zone parameters presented in the next section is identified from measurements of the toughness at the onset of crack propagation, under quasi-static conditions. Dynamic fracture is not investigated so that a comparison with results or conclusions found in Bjerke and Lambros [22] or in Murphy and Ivankovic [23] is not straightforward. However, dynamic loading will be investigated in a forthcoming study. 4. Calibration of the cohesive zone representing crazing Since the condition for craze breakdown has been extensively investigated by interferometry with numerous data collected and performed by Do¨ll and Ko¨nczo¨l [24,25], we borrow from these studies the value of the critcr ical craze thickness Dcr n for the nucleation of a crack in PMMA, with Dn ¼ 3 lm. Therefore, we have focused on the characterization of the first two stages of crazing, namely initiation and thickening. The material under consideration is a commercial PMMA (Perspex) supplied in the form of plates with a thickness of 10 mm. The specimens used for the analysis of craze initiation and craze thickening are from the same plate. The molecular weight measured by size exclusion chromatography results in Mn = 864 kg/mol and MW = 1843 kg/mol. Do¨ll [24] has reported an influence of the molecular weight on the development of a stable craze. For PMMA, a critical value about 200 kg/mol is evidenced so that the mechanism of crazing depicted by Kramer [2] is expected to operate for our material. The principal and secondary relaxations were charac-

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terized by dynamic mechanical spectroscopy. The glass transition identified as the principal relaxation at 1 Hz occurs at 400 K while the b relaxation appears at 300 K, the room temperature. 4.1. Craze initiation The experiment used to investigate craze initiation is similar to that of Sternstein et al. [26]. It consists of a plate with a circular hole subjected to a remote constant tensile stress (see Fig. 4a). The loading increases up to r1 during one minute and is then maintained at a constant value. The crazes initiate at the equator of the hole and extend over a finite region. The optical observation of the crazed region shows a zone filled with crazes while the other is not (Fig. 4b). The frontier between these two zones defines the local critical stress state for craze initiation. The contour of the region with crazes is observed to be stable for a loading time larger than 20 min for a remote stress of 30 MPa, at room temperature. By embedding cohesive zones across the volume, Tijssens et al. [4] have shown that multiple crazing as observed here does not noticeably affect the elastic stress distribution derived from the problem of a plate with a hole (without crazes) [27], as long as the crazed region does not exceed q/2, with q the hole radius. By taking various points regularly picked up along the craze frontier, we estimate the local principal stresses r1 and r2 from the elastic solution. These are reported in Fig. 4c in the principal stress space (circles). We observe that the craze frontier corresponds to a limited domain in the stress space (r1, r2) with r1 = 55 MPa ± 2 MPa and r2 = 7 MPa ± 3 MPa at the boundary of the crazed region. In the same plot, we have reported the experimental data extracted from studies of craze initiation due to Sternstein et al. [17,20] and Oxborough and Bowden [18] (obtained for PMMA at 60 C and PS at room temperature respectively). In the first stress quadrant (r1 > 0 and r2 > 0), we observe that craze initiation takes place for a maximum principal stress rmax approx1 imately constant. Thus, as a first approximation and for the domain (r1 > 0; r2 > 0), we consider a craze initiation criterion based on maximum principal stress with rcr 1;max ¼ 55 MPa. This value is obtained by averaging our experimental data and is indicated by a dashed line in Fig. 4c.

0.1 mm Craze frontier

σ

σ

Hole ρ=0.2 mm

(b)

(a) 100

σ 1 (MPa)

80

[18]

60

[20] T = 60°C Adopted initiation criterion

40

[17] T = 60°C Our experimental data

20

σ 2 (MPa)

0 -100

-60

-20

20

60

100

(c) Fig. 4. (a) Schematic description of the experiment to analyze craze initiation, (b) optical observation of the region with crazes and (c) value of the stresses for craze initiation reported in the principal stress space and adopted criterion based on a local maximum principal stress initiation criterion.

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4.2. Craze thickening and related energy release rate Since crazing is the mechanism responsible for failure with this process being assumed to be viscoplastic, the material toughness is expected to be time dependent even if the bulk response is elastic (or slightly viscoelastic). The variation of the toughness with the loading rate is one of the key features for the calibration of the parameters used in the description of the craze thickening rate. We present the experimental analysis of the toughness for various loading conditions, which are then used for the craze parameter identification. 4.2.1. Specimen preparation According to the standard protocols [28–30], sharp notches are introduced following two steps. First, a prenotch of 0.25 mm radius is mill cut with a circular cutting wheel. In order to prevent heating while machining, this operation is done in an automatic mode in the presence of fresh compressed air. A sharp notch is further introduced at the tip of the pre-notch by tapping a razor blade. A mass (200 g) is dropped from a height of 18 cm through a stem fixed on the device. This runner impacts the razor blade placed in the throat of the prenotch. The procedure results in reproducible sharp cracks, with a length at least four times larger than the radius of the pre-notch (in agreement with the recommendation of the ESIS-TC4 [30]). An example of a sharp notch observed under crossed polarisers is shown in Fig. 5. It is seen that the procedure of tapping induces few initial stresses apart from a small zone at the end of the sharp crack. 4.2.2. Experimental set-up Single notch specimens are loaded under a four-point bending configuration to ensure mode I (see Fig. 6). The variation of the toughness with the loading rate is then investigated. Fracture tests are carried out using an INSTRON servo hydraulic tensile test machine in which a force rate is prescribed thus corresponding to a constant rate for the stress intensity factor (SIF) K_ I ¼ dK I =dt with [31] pffiffiffiffiffiffi ð9Þ K I ¼ r paF ða=W Þ:

Fig. 5. Optical observation of a sharp crack prepared by tapping in PMMA.

S1 P/2

P/2

W a

S2 Fig. 6. Four-point bending configuration for mode I test: thickness B = 10 mm, width W = 20 mm, crack length a with 0.45 < a/W < 0.55, S1 = 90 mm, S2 = 40 mm.

N. Saad-Gouider et al. / Engineering Fracture Mechanics 73 (2006) 2503–2522 400

400

F (N)

F(N)

K I = 10−2MPa. m/s

K I = 10−3MPa . m/s 300

300

200

200

100

2513

100 u (mm)

u (mm) 0

0 0

400

0.2

0.4

0.6

0.8

1

0 400

F (N)

0.2

0.4

0.6

0.8

1

F(N)

K I = 10−1 MPa. m/s

K I = 1MPa . m/s

300

300

200

200

100

100 u (mm)

0

u(mm) 0

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Fig. 7. Force versus displacement curves recorded for various loading rates and sharp notched specimens.

In Eq. (9), a is the crack length and W the specimen width. The geometrical factor for the four-point bending configuration is [31] 2

3

4

F ða=W Þ ¼ 1:112  1:4ða=W Þ þ 7:33ða=W Þ  12:08ða=W Þ þ 14ða=W Þ : The specimens pffiffiffiffi are tested under pffiffiffiffi ambient conditions (temperature and humidity) for loading rates ranging from 103 MPa m=s to 1 MPa m=s. As observed from the force versus displacement curves (Fig. 7), the onset of crack propagation occurs when a maximum force is attained, thus defining the critical force and related toughness. 4.2.3. Variation of toughness with loading rate The variation of the toughness with loading rate is reported in Fig. 8. The size requirements for plane strain and small scale yielding conditions

Fig. 8. Variation of the critical KIC with loading rate for specimens with an initial sharp notch.

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a; B; ðW  aÞ P 2:5

 2 K IC ; ry

ð10Þ

in which B is the specimen thickness, have been validated for a  10 mm, W = 20 mm and B = 10 mm. The yield stress ry corresponds to that for uniaxial tension and following the ESIS-TC4 requirement [30], its value has to be estimated for a time scale comparable to that of the fracture test. For a given loading rate, a relevant time scale for the loading can be defined through tfailure ¼ K IC =K_ I . The characteristic time scale ty is derived from the ratio ry =r_ or ey =_e which are estimated from the simulations of uniaxial tensile tests at various strain rates, with the parameters identified for the bulk in Section 2. From these numerical data, the relationship between the yield stress ry with ty results in a linear variation. In order to estimate the relevant plastic threshold for a given fracture test, we adopt ty = tfailure as a characteristic time scale. The related value of ry derived from our numerical data is identified and used for the check of the size requirements (10). The yield stress is pffiffiffiffi found p toffiffiffiffi vary from 66 MPa to 111 MPa for loading rates ranging from K_ I ¼ 103 MPa m=s to 1 MPa m=s. Small scale yielding conditions are verified together with plane strain conditions for all the experimental points reported in Fig. 8. Plane strain conditions were also confirmed from the observation of the fracture surfaces with a straight crack front profile, which appears normal to the direction of crack propagation. At the onset of crack advance, slow crack growth precedes unstable crack propagation. The maximum force corresponds to a plateau observed in the records reported in Fig. 7. This highest force is used for the calculations of the critical SIF KIC. In Fig. 8, we observe that the critical toughness KIC increases about 30% for a loading rate varying over four decades. 4.2.4. Estimate of the energy release rate The increase of KIC observed with increasing loading rate K_ I is noticeable but not sufficient to demonstrate that failure is rate dependent. The viscoelastic effects need to be accounted for to derive the energy release rate from GIC ¼ K 2IC =E0 ;

ð11Þ 2

in which E 0 = E/(1  m ) for plane strain conditions, E a Young’s modulus to be defined and m the Poisson’s ratio. The material under consideration exhibits some viscoelasticity which has been represented by a secant Young’s modulus for the large deformation description of the bulk response in Section 2. In the present mode I fracture tests, the yield stress is not attained and the viscoelastic effects are reduced. The estimate of the viscoelasticity is derived from the measure of the secant modulus of un-notched specimens loaded for stress rate r_ identical to those of the fracture tests up to a load level rmax for which failure is observed on the notched samples. In the bending tests performed on the un-notched specimens, we use an LVDT transducer to measure the maximum deflection u of the beam loaded up to rmax. The secant modulus is then derived from 1 Esecant ¼ F max ðS 1  S 2 Þð2S 21 þ 2S 1 S 2  S 22 Þ, with S1 and S2 indicated in Fig. 6. The secant u 8BW 3 pffiffiffiffi modulus evolves pffiffiffiffi secant from EEXP ¼ 3:22 GPa to 3.82 GPa for a loading rate K_ I varying from 1  103 MPa m=s to 1 MPa m=s, showing that viscoelastic effects are not negligible. By accounting for the time dependency of this secant modulus and by assuming that the Poisson ratio remains constant, the variation of the critical energy release rate GIC with loading rate is reported in Fig. 9. We do observe an increase of 35% of GIC with loading rate which evidences the rate dependent nature of the failure process. In the following section, we show how the parameters involved in the craze thickening kinetics (10) are derived from the variation of the toughness GIC with loading rate. 4.3. Calibration of the craze thickening kinetics The aim of this section is to identify the parameters ðD_ 0 ; Ac ; rc Þ of the craze thickening rate (6). The calibration is based on the comparison of numerical predictions of the energy release rate with the experimental results reported in Fig. 9. The simulations are performed with a finite element analysis of mode I failure depicted in Fig. 10 and detailed in [1]. Plasticity and crazing are assumed to be confined around the crack tip so that the small scale yielding framework is allowed. The boundary layer approach is used to investigate

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Fig. 9. Variation of the energy release rate GIC with loading rate for samples with an initial sharp notch.

Fig. 10. (a) Description of the mode I small scale yielding problem, (b) the mesh and (c) zoom of the mesh around the crack.

the mode I plane strain conditions. A cohesive surface is laid out ahead of the crack along the symmetry plane, where craze initiation is most probable and a single cohesive zone (a single craze) is considered, as observed experimentally. The remote region consists in a circular arc along which the KI displacement fields are prescribed [1]. The constitutive law for the bulk material is elastic–viscoplastic and not restricted to a linear elastic response but shear yielding and crazing are allowed as in [1]. The framework presented in [1] is used with a total Lagrangian finite deformation description of the deformation and the rate form of the virtual work to solve the problem incrementally. In order to illustrate the main steps of a simulation, we pffiffiffiffihave reported in Fig. 11 the distribution of the plastic shear strain rate c_ p for the loading rate K_ I ¼ 1 MPa m=s and the set ðD_ 0 ; rc ; Ac Þ identified in the next parpffiffiffiffi agraph. The variable c_ p is conveniently normalized by C_ 0 ¼ K_ I =ðs0 rt Þ which represents a prescribed shear strain rate at rt (rt being the crack tip radius, with rt = 5 lm in the calculations), s0 being the athermal yield stress for bulk viscoplasticity (see Section 2). In Fig. 11a–c, the corresponding contours for (a) at the onset of craze initiation, (b) during craze thickening and prior to the first craze fibril breakdown, (c) during crack

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4

y/rt

4

3 2

location of craze initiation

y/rt

4

3

3

2

2

1

0

0

-1

-1

-1

-2

-2

-2

-3

-3

x/rt 0

1

2

3

4

5

6

7

-4 -1

. . γ p/γ 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

1

-4 -1

y/rt

0 crack

craze

x/rt 0

1

(a)

2

3

4

5

6

7

-3 -4 -1

x/rt 0

1

2

(b)

3

4

5

6

7

(c)

p

Fig. 11. Distribution of the plastic shear strain rate c_ : (a) at craze initiation, (b) prior to the onset of craze fibril breakdown and (c) during pffiffiffiffi crack propagation ðK_ I ¼ 1 MPa m=sÞ.

0.15

KI / E

secant

rt 0.1 KI = 1MPa m/s

craze initiation

KI = 1×10−1 MPa m/s

0.05

crack propagation

K I = 1×10−2 MPa m/s KI = 1×10−3 MPa m/s

length (craze+crack)/rt

0 0

2

4

6

8

10

Fig. 12. Resistance curve in term of KI versus the length of the (craze + crack) corresponding to the distribution in Fig. 11 for pffiffiffiffi K_ I ¼ 1 MPa m=s and other loading rates.

propagation are presented. In this case, the bulk response shows negligible plasticity as c_ p =C_ 0  0 and is primary elastic. pffiffiffiffi In Fig 12, we report the variation of the load level KI (actually the normalized quantity K I =Esecant rt ) versus the length of the craze plus crack. Once crazing initiates (circle in Fig. 12), craze thickening results in a ductile response with an increasing load required for further fibrillation. At the onset of craze fibril breakdown (square in Fig. 12), crack propagation takes place for a constant load level corresponding to a plateau value for KI which is used for the definition of the critical KIC in the simulations. The related energy release rate is extracted by using (11) with the modulus Esecant = 2.2 GPa identified in the compression tests (Section 2). The predictions are then compared to the experimental data for the calibration. The simulations predict a craze length about 30–40 lm which is comparable to the measures reported by Do¨ll and Ko¨nczo¨l [24,25]. By using the maximum principal stress rcr 1 ¼ 55 MPa for craze initiation and a critical craze thickness cr Dn ¼ 3 lm for the craze fibril breakdown, the experimental data reported in Fig. 9 are then used to calibrate the parameters involved in the craze thickening rate ðD_ 0 ; rc ; Ac Þ. The procedure consists in fixing Ac first and then to identify ðD_ 0 ; rc Þ by fitting the numerical predictions with the experimental data. The value of Ac is chosen by comparing the temperature dependence of the craze stress during craze thickening with that of the bulk yield stress. We report in Fig. 13 the variation of craze stress rn with temperature extracted from Do¨ll’s and Ko¨nczo¨l’s reviews [24,25], and that of the yield stress estimated from the simulation of the bulk response at a strain rate e_ ¼ 103 s1 at various temperatures. We observe a temperature dependence similar for the craze stress and the yield stress. Based on this observation, we consider an identical temperature dependence for the craze thickening rate and for the bulk viscoplasticity with Ac  A = 60 K/MPa. With the value of the temperature dependence thus fixed, the couple of parameters ðD_ 0 ; rc Þ is adjusted to ensure that the energy release rate GIC predicted from the simulation of the mode I configuration matches with the experimental data of Fig. 9.

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140

140

120

120

100

100

80

80

60

60

40

40

σ n (MPa)

160

20

σ y (MPa)

160

2517

20

T (°C) 0

0

0

20

40

60

80

100

120

Fig. 13. Variation of the craze stress rn [24,25] and the yield stress ry with temperature derived from simulations of a uniaxial tensile tests at various strain rates.

550

GIC (J/m2 )

450 .

σ c = 110 MPa, Δ0 = 10 −4 mm/s . σ c = 140. MPa,Δ0 = 10−1 mm/s . σ c = 150 MPa, Δ0 = 1mm/s . σ c = 160 MPa, Δ0 = 10mm/s . σ c = 180 MPa, Δ0 = 100mm/s

350

Experimental data

250 -4

-3

-2 . -1 Log(KI ) (MPa. m /s)

0

1

Fig. 14. Variation of the energy release rate with loading rate for different parameters of the cohesive zone which fit the experimental data.

Following a trial and error procedure, we identify the pair ðD_ 0 ; rc Þ ranging from (104 mm/s, 110 MPa) to (100 mm/s, 180 MPa) of which predictions of GIC are within the experimental scatter (see Fig. 14). The continuous curve reported in Fig. 14 corresponds to the couple of parameters ðD_ 0 ; rc Þ ¼ ð0:1 mm=s;140 MPaÞ which best captures the variations of GIC with loading rate. 5. Validation In order to validate the set of parameters calibrated for PMMA, we compare experiments with predictions of the toughness of two other four-point bending configurations in which blunted notches are used instead of a sharp notch. Both the size effect introduced by a blunted notch of two different radii and the loading rate are investigated. We first report the measurements of GIC with loading rate which are going to be compared to the simulations. We repeat the procedure of deriving the energy release rate from the measure of the critical toughness and secant modulus representing the viscoelastic effects for a given loading rate. Those measurements of Gc are reported in Fig. 15. The level of the toughness for a crack tip radius of 0.25 mm and 0.5 mm is five to ten times larger when compared to that reported for sharp cracks. The observation of the crack tip zone after failure shows multiple crazing along the notch tip (Fig. 16a) while a single craze has developed when a sharp crack was used. In Fig. 16b, a closer observation indicates that crazes appear to be regularly spaced along the notch contour and does not correspond to diffuse crazing as observed in the analysis of initiation (Fig. 4b). We believe that the presence of surface defects introduced when machining the notch triggers multiple crazes.

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GIC(J/m 2 ) rt = 500μm

4000 3000

rt = 250μm

2000 1000

sharp crack . Log (K I) (MPa. .m /s)

0 -4

-3

-2

-1

0

1

Fig. 15. Variation of the energy release rate with loading rate in the case of sharp and blunted notches rt.

Fig. 16. (a) Observation of multiple crazes around a blunt notch in PMMA, (b) closer observation of the crazes along the notch and (c) trajectories for allowed crazing accounted for in the simulation.

In our simulations, we account for this ‘‘extrinsic effect’’ by authorizing multiple trajectories for the crazes, as observed experimentally and indicated in Fig. 16c. We insert five cohesive zones regularly distributed on both sides of the symmetry plane of the notch which are separated by an angle of 7.5 as suggested by the experimental observation. The predictions of the simulation are then compared to the experimental results (curves in Fig. 15). We observe that the size effect of the notch radius and the variation of the energy release rate with loading rate are correctly predicted. This comparison shows that the description is able to produce quantitative results and confirms the need to use a viscoplastic cohesive zone for crazing. The blunted configurations analyzed here is symmetric with craze oriented towards an inclined direction from the crack tip. As the crack advances, the propagation takes place along the crack symmetry plane thus relaxing the stress where the fibrils have broken down and causing a stress relaxation on the crazes which have developed out of this plane. Propagation takes place for a constant load level with most of the deformation being carried out by the cohesive zone at the crack symmetry plane. No branching is considered and details on the crack propagation beyond the early stages of crack propagation are not presented in the present study but could be included in a forthcoming study in which the entire specimen will be analyzed. 6. Parameters influencing the calibration With the set of the cohesive zone parameters calibrated in the foregoing sections, we now investigate the influence of the craze critical thickness which is not identified but borrowed from [24,25] and also that of the craze initiation condition on the overall craze process and related failure. 6.1. Craze initiation condition In order to investigate the effect of the stress level rcr 1 for craze initiation, we vary the identified value of ¼ 55 MPa from 1 MPa to 100 MPa. A value of the critical stress equal to one corresponds to a situation in which the craze kinetics only govern the mechanism of crazing while a value of 100 MPa would indicate that rcr 1

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GIC (J/m2)

500

400

σ1cr = 55MPa σ1cr = 100MPa σ1cr = 150MPa

300

Experimental data

200 -4

-3

-2 -1 . Log (KI) (MPa. m/s)

0

1

Fig. 17. Variation of the energy release rate with loading rate for various stress level for craze initiation.

craze nucleation is retarded when compared to the calibrated value (55 MPa). We comparepthe ffiffiffiffi simulations and the predictions of the toughness obtained for loading rate as 1  103 6 K_ I 6 1 MPa m=s. We have observed that changing the value of the stress level for craze initiation does not affect noticeably the predicted toughness as long as rcr 1 6 100 MPa (see Fig. 17), since the value of the normal stress rn at which craze thickening takes place is about 100–110 MPa. Therefore, the traction-opening curve for crazing we identify for PMMA follows a hardening-like trajectory similar to that depicted (2a) n Fig. 3. For a larger value of rcr 1 , for instance for rcr ¼ 150 MPa, the predicted toughness is not in agreement with the experimental values 1 pffiffiffiffi for K_ I ¼ 1  103 MPa m=s. In this case, a larger value of KIC is predicted due to the development of some plasticity of the bulk near the crack tip which results in a more ductile response, which disagrees with the experimental observations. This is not seen for higher loading rates. Therefore as long as rcr 1 6 100 MPa, we evidence a minor influence of the condition for craze initiation in our cohesive zone description, and for the loading rates investigated here. Such a difference between the normal traction at craze initiation (55 MPa) and that for fibrillation (about 100–110 MPa) is not consistent with Kramer’s description [21,2] in which the thickening of the fibrils is observed right after craze initiation. The discrepancy probably originates from the use of two experimental set-up to estimate the condition for craze initiation and the analysis of the craze thickening through the fracture tests. The present analysis of the influence of craze initiation indicates that for the loading conditions under consideration here, the traction for craze initiation cannot be larger than that for fibrillation so that a hardeninglike response as (2a) in Fig. 3 is identified for PMMA. 6.2. Craze critical thickness As numerous and consistent measurements of the craze critical thickness Dcr n are available in the literature [24,25], we did not focus on the identification of this parameters for the material we used but borrowed the value of Dcr n ¼ 3 lm reported for PMMA at room temperature. We now estimate the influence of this parameter on our toughness predictions at various loading rates, for a configuration with an initial sharp crack. To this end, cr we consider a variation of Dcr n about ±0.5 lm (±16%) with respect to the reference value ðDn ¼ 3 lmÞ, all the other parameters being taken from the calibration presented in Table 2. In Fig. 18, we have reported the Table 2 Calibrated cohesive zone parameters for the description of crazing in PMMA Cohesive zone parameters

Initiation

Thickening

rcr 1 ðMPaÞ

Ac (K/MPa)

ðD_ 0 ðmm=sÞ; rc ðMPaÞÞ

Breakdown Dcr n ðlmÞ

Validated range

(1–100)

60

3

Adopted value

55

60

(104 mm/s, 110 MPa)– (100 mm/s, 180 MPa) 0.1

140

3

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GIC (J/m 2 ) Δ crn = 3.5µm

500

reference : Δ crn = 3µm Δ crn = 2.5µm

400

300

Experimental data 200 -4

-3

-2

-1

. Log ( K I )(MPa. m/s)

0

1

Fig. 18. Influence of the craze critical thickness on the variation of energy release rate with loading rate.

predictions of the energy release rate GIC for various loading rates and the sharp crack configuration. For a given loading rate, we observe that an increase (respectively a decrease) of the craze critical thickness Dcr n results in crack propagation for a higher (respectively lower) material toughness. There is a linear correlation between the value of the energy release rate and the variations of Dcr n . For a given loading rate, the value of the normal traction rn during the thickening of the craze is the same while a variation of the critical opening Dcr n results in an cr energy release rate approximately equal to GIC  rn  Dcr , which scales with D as observed in the simulations. n n Therefore, it appears that the parameter Dcr is of major importance in the calibration procedure. Its value has to n be measured carefully to allow accurate predictions of the material toughness. Such measurement can be performed conveniently by interferometry as reported by Do¨ll and Ko¨nczo¨l [24,25]. 7. Conclusion A coupled experimental and numerical analysis of crazing in PMMA is presented, in which the detailed protocol for the calibration of a viscoplastic cohesive zone for crazing is presented, together with the identification of the parameters involved in the elastic–viscoplastic constitutive law for the bulk. The calibration is then validated by comparing the predictions and measurements of the toughness at different loading rates with two different notch radii. Both size and rate effects are captured accurately by the model and the following major points arise: • The rate dependency of the toughness and energy release rate is evidenced, for quasi-static conditions and tests performed at room temperature. The viscoelastic effects are accounted for in the estimate of the energy release rate which is observed to increase about 35% for a loading rate varying over four decades. • The material under consideration is a commercial PMMA for which no plasticity at the crack tip is observed in the fracture tests, its bulk response being elastic (slightly viscoelastic) for the loading rates and the mode I configuration analyzed in the present study. The variation of the energy release rate GIC is due to the craze viscoplastic response. The toughness is then governed by the ratio between the time scale related to crazing tc with respect to that related to the loading tload. It is demonstrated that a viscoplastic cohesive zone formulation is necessary to predict accurately the toughness at any rate. The use of a rate independent cohesive zone would correspond to some average or approximate estimate. • The shape of the traction-opening law identified shows a transient hardening response followed by a plateau. This response corresponds to the description within quasipffiffiffiffi of crazing in PMMA, at room temperature pffiffiffiffi static and slow loading rates (K_ I 6 1 MPa m=s and KIC of the order of 1 MPa m). The last point may seem in contradiction with other conclusions for the formulation of the traction-opening law found in Bjerke and Lambros [22] or Murphy and Ivankovic [23] in which a linear decay of the normal traction up to Dcr or a softening followed by a plateau up to Dcr of rn are identified. Those studies have focused on dynamic fracture, and differences in the critical stress for crazing with loading times smaller than 1 ms may explain the different conclusions derived in these studies.

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The analysis of higher loading rates with fast tensile tests at 1–10 m/s and dynamic fracture will be the purpose of forthcoming studies, in order to check the validity of the present calibration. The investigation of crazing with the present viscoplastic cohesive zone for more ductile glassy polymers like polycarbonate will also be considered. Acknowledgements The computation reported here were carried out at the Centre Informatique National de l’Enseignement Supe´rieur at Montpellier-France under the grant c2005 09 22721. Access to this facility and technical support from CINES are acknowledged by NS and RE. The ultra-sonic measurements of the elastic wave velocities and related derivation of the elastic constants were carried out by Ph. Guy from GEMPPM who is gratefully acknowledged by RE. References [1] Estevez R, Tijssens MGA, Van der Giessen E. Modelling of the competition between shear yielding and crazing in glassy polymers. J Mech Phys Solids 2000;48:2585–617. [2] Kramer EJ. Microscopic and molecular fundamentals of crazing. Adv Polym Sci 1983;52–53:1–56. [3] Kramer EJ, Berger LL. Fundamental processes of craze growth and fracture. Adv Polym Sci 1990;91–92:1–68. [4] Tijssens MGA, Van der Giessen E, Sluys LJ. Modelling of crazing using a cohesive surface methodology. Mech Mater 2000;32:19–35. [5] Van der Giessen E, Lai J. A numerical study of craze growth. Deformation, yield and fracture of polymers, vol. 10. London: The Institute of Materials; 1997. p. 35–8. [6] Ishikawa M, Narisawa I, Ogawa H. Criterion for craze nucleation in polycarbonate. J Polym Sci 1977;15:1791–804. [7] Boyce MC, Parks DM, Argon AS. Large inelastic deformation of glassy polymers, part I: Rate dependent constitutive model. Mech Mater 1988;7:15–33. [8] Wu PD, Van der Giessen E. On improved network models for rubber elasticity and their applications to orientation hardening in glassy polymers. J Mech Phys Solids 1993;41:427–56. [9] Wu PD, Van der Giessen E. Computational aspects of localized deformations in amorphous glassy polymers. Euro J Mech, A/Solids 1996;15:799–823. [10] Argon AS. A theory for the low-temperature plastic deformation of glassy polymers. Philos Mag 1973;28:839–65. [11] Arruda EM, Boyce MC. A three dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys Solids 1993;14(2):389–411. [12] Bowden PB. The yield behaviour of glassy polymers. In: Haward RN, editor. The physics of glassy polymers. London: Applied science publishers; 1973. p. 279–339. [13] Basu S, Van der Giessen E. A thermo-mechanical study of mode I, small-scale yielding crack-tip fields in glassy polymers. Int J Plast 2002;18:1395–423. [14] Arruda EM, Boyce MC, Jayachandran R. Effects of strain rate, temperature and thermo mechanical coupling on the finite strain deformation of glassy polymers. Mech Mater 1995;19:193–211. [15] Estevez R, Basu S, Van der Giessen E. Analysis of temperature effects near mode I cracks in glassy polymers. Int J Fract 2005;122(3):249–73. [16] Kausch HH. Polymer fracture. 2nd ed. Heidelberg: Springer; 1987. [17] Sternstein SS, Ongchin L. Yield criteria for plastic deformation of glassy high polymers in general stress fields. Polym Prep Am Chem Soc Polym Chem 1969;10:1114–24. [18] Oxborough RJ, Bowden PB. A general critical-strain criterion for crazing in amorphous glassy polymers. Philos Mag 1973;28:547–59. [19] Argon AS, Hannoosh JG. Initiation of crazes in polystyrene. Philos Mag 1977;36:1195–216. [20] Sternstein SS, Myers FA. Yielding of glassy polymers in the second quadrant of principal stress space. J Macromol Sci Phys B 1973;8:539–71. [21] Donald AM, Kramer EJ. The mechanism of craze tip advance in glassy polymers. Philos Mag A 1981;43:857–70. [22] Bjerke TW, Lambros J. Theoretical development and experimental validation of a thermally dissipative cohesive zone model for dynamic fracture of amorphous polymers. J Mech Phys Sol 2003;51:1147–70. [23] Murphy N, Ivankovic A. The prediction of dynamic fracture evolution in PMMA using a cohesive zone model. Engng Fract Mech 2005;72:861–75. [24] Do¨ll W. Optical interference measurements and fracture mechanics analysis of crack tip craze zones. Adv Polym Sci 1983;52– 53:105–68. [25] Do¨ll W, Ko¨nczo¨l L. Micromechanics of fracture: optical interferometry of crack tip craze zone. Adv Polym Sci 1990;91–92:128–214. [26] Sternstein SS, Ongchin L, Silverman A. Inhomogeneous deformation and yielding of glasslike high polymers. Appl Polym Sympos 1968;7:175–99. [27] Timoshenko S, Goodier JN. Theory of elasticity. New York: McGraw-Hill; 1961.

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[28] De´termination de la te´nacite´ a` la rupture (GIC et KIC) Application de la me´canique line´aire e´lastique de la rupture (LEFM). ISO 13586, ISO, Gene`ve, 2000. [29] Standard test methods for plane-strain fracture toughness and strain energy release rate of plastic materials. ASTM 5045, 1999. [30] Williams JG, Pavan A. Fracture of polymers, composites and adhesives II. Amsterdam: Elsevier; 2003. p. 578. [31] Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. London: Professional Engineering Publishing; 2000.