Mechanics of dynamic fracture in notched polycarbonate

Author’s Accepted Manuscript Mechanics of Dynamic Fracture in Notched Polycarbonate Anshul Faye, Venkitanarayanan Parmeswaran, Sumit Basu www.elsevier.com/locate/jmps

PII: DOI: Reference:

S0022-5096(15)00004-6 http://dx.doi.org/10.1016/j.jmps.2015.01.003 MPS2581

To appear in: Journal of the Mechanics and Physics of Solids Received date: 24 September 2014 Revised date: 26 November 2014 Accepted date: 6 January 2015 Cite this article as: Anshul Faye, Venkitanarayanan Parmeswaran and Sumit Basu, Mechanics of Dynamic Fracture in Notched Polycarbonate, Journal of the Mechanics and Physics of Solids, http://dx.doi.org/10.1016/j.jmps.2015.01.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Mechanics of Dynamic Fracture in Notched Polycarbonate Anshul Fayea , Venkitanarayanan Parmeswarana , Sumit Basua,1,∗ a Department

of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, UP India.

Abstract Fracture toughness of brittle amorphous polymers (e.g. polymethyl methacrylate(PMMA)) has been reported to decrease with loading rate at moderate rates and increase abruptly thereafter to close to 5 times the static value at very high loading rates. Dynamic fracture toughness that is much higher than the static values has attractive technological possibilities. However, the reasons for the sharp increase remain unclear. Motivated by these observations, the present work focuses on the dynamic fracture behavior of Polycarbonate (PC), which is also an amorphous polymer but unlike PMMA, is ductile at room temperature. Towards this end, a combined experimental and numerical approach is adopted. Dynamic fracture experiments at various loading rates are conducted on single edge notched (SEN) specimens with a notch of radius 150 µm, using a Hopkinson bar setup equipped with ultra high-speed imaging (> 105 fps) for real-time observation of dynamic processes during fracture. Concurrently, 3D dynamic finite element simulations are performed using a well calibrated material model for PC. Experimentally, we were able to clearly capture the intricate details of the process, for both slowly and dynamically loaded samples, of damage nucleation and growth ahead of the notch tip followed by unstable crack propagation. These observations coupled with fractography and computer simulations led us to conclude that in PC, the fracture toughness remains invariant with loading ˙ from static rate at Jfrac = 12 ± 3 kN/m for the entire range of loading rates (J) to 1 × 106 kN/m − s. However, the damage initiation toughness is significantly higher in dynamic loading compared to static situations. In dynamic situations, damage nucleation is quickly followed by initiation of radial crazes from around the void periphery that initiate and quickly bridge the ligament between the initial damaged region and the notch. Thus for PC, two criteria for two major stages in the failure process emerge. Firstly, a mean stress based defect initiation is suggested. The value of the critical mean stress for defect initiation under dynamic loading is found to be 115±5 MPa, which is significantly higher than its static value of 80 MPa. The critical normal plastic stretch needed for crazes to nucleate from the nucleated defect is estimated to be about 1.78 ± 0.2. ∗ Corresponding

author. Tel.: +91 5122597506; Fax: +91 5122597408. Email address: [email protected] (Sumit Basu)

Preprint submitted to Elsevier

January 13, 2015

Keywords: Polycarbonate; Dynamic fracture; Finite Element Simulation; Fracture toughness; Craze; Hopkinson Bar; DIC

1. Introduction At very high rates of loading, steep increase in fracture toughness over values under static conditions have been reported for metals, alloys, polymers, ceramic and metallic glasses [Belenky et al. (2010), Chichili and Ramesh (1995), Koyabashi et al. (1988), Ravi-Chander and Knauss (1984), Rittel and Rosakis (2005), Weerasooriya et al. (2006), Yokoyama (1995)]. For the amorphous brittle polymer polymethyl methacrylate (PMMA) for example, fracture toughness √ at loading rates of ∼ 105 MPa ms−1 has been reported to be about 5 times that under static conditions [Huang et al. (2009), Rittel and Maigre (1996), Wada (1992), Wada et al. (1993), Zhou et al. (2006)]. We have summarised the experimentally obtained variation of the dynamic fracture initiation toughness KId with the loading rate K˙ Id in Fig. 1. Different sample geometries, loading and methods for calculation of the fracture toughness have been used by various investigators. For example, Wada et al. (1993) used a hybrid experimental and numerical approach where, impact fracture tests on single edge notched (SEN) specimens were performed using one point bend loading, following which, from Finite Element (FE) analyses, stress intensity factors (SIF) were calculated using crack tip opening displacement (CTOD). In a completely experimental approach, Rittel and Maigre (1996) used compact tension (CT) specimens loaded using a split-Hopkinson pressure bar (SHPB). They used the H-integral to calculate the dynamic SIF. On the other hand, Zhou et al. (2006) used Brazillian disk specimens, loaded in SHPB and calculated the dynamic SIF by using the dynamic fracture load in the static SIF formula for this specimen. Finally, Huang et al. (2009) used semicircular disk specimens in SHPB and, like Zhou et al. (2006), used the dynamic fracture load in the corresponding static SIF formula. From the wide range of methods used, it seems that enhancement of fracture toughness at high rates of loading is an intrinsic behaviour of the material. Apart from being an intriguing phenomenon that is not clearly understood, this has important technological implications. Components that are expected to be subjected to low static loads but can experience impact like loading can be made much lighter and smaller if fracture initiation toughness indeed is high under high rates as observed in these experimental results. Ravi-Chander and Yang (1997), in an experimental study of several materials, concluded that the mechanism governing fracture in brittle materials involves the nucleation, growth and coalescence of microcracks. Microcracking ahead of a propagating crack leads to characteristic patterns on the fracture surface and also to a constant crack speed. In an attempt to explain the behavior of increased fracture toughness at high loading rates in PMMA, Estevez and Basu (2008) performed a series of ‘what if’ simulations that looked at different

2

scenarios capable of explaining all experimental observations on high loading rate behaviour of these materials. The two most important observations that they sought to reconcile were those of substantial thermo-elastic cooling at crack tips [Rittel (1998)] and the enhancement of fracture toughness at high loading rates. They concluded that for both these observations to be correct, supression of plasticity and a drastic change in the failure mechanism at high loading rates is necessary. In fact, to explain the enhancement in dynamic SIF, it was estimated that at high rates of loading, the fracture mechanism in PMMA has to change to one that needs an opening stress of almost 1 GPa to initiate. Polycarbonate (PC) is a more ductile polymer compared to PMMA. Moreover, the impact resistance of PC, as measured from Izod impact tests [Allen et al. (1973), Fraser and Ward (1977), Inberg and Gaymens (2002), Ravetti et al. (1975)] , is among the highest for thermoplastics [Casiraghi (1978)], making it suitable for high impact applications. Indeed, in a number of common applications of PC in automobile, aerospace and defense industries, it is routinely subjected to strain rates above 1 × 103 s−1 [Shah (2009), Wright et al. (1993)]. Thus it is interesting and important to see if PC also undergoes large enhancements in its resistance to fracture at high rates. An in-depth investigation of the fracture behaviour of PC has three major challenges. Evidently, we need to be able to measure the fracture toughness KI of the material√reliably at loading rates ranging from quasi-static to about K˙ I ∼ 1×105 MPa ms−1 . Secondly, at very high rates, suppression of plasticity near the tip of a notch or defect is expected. But for a ductile material like PC, complete absence of plasticity is unlikely. Therefore, we need to closely observe the development of plasticity close to the notch tip. Thirdly, the evolution and nature of damage at the tip of a notch needs to be monitored and quantified. All these are compounded by the fact that the entire process of damage evolution and failure occurs within a few microseconds and affects a very small region around the tip of the notch. The above challenges are handled using a variety of experimental and numerical means. Notched 3 point bend (3PB) specimens of PC have been dynamically loaded using a Hopkinson pressure bar (HPB). Ultra high speed imaging is used to accurately ascertain the time to initiation of failure. The failure time and the load point displacement history serves as essential inputs to a FE code. Dynamic, large deformation based FE simulations are run with an elastoviscoplastic constitutive model that has been calibrated carefully under uniaxial situation [Kattekola et al. (2014)] for the material in question. The dynamic J integral, calculated numerically at the time of fracture gives the toughness at crack initiation. Displacement fields obtained through Digital Image Correlation (DIC) performed around the tip of the notch in the HPB experiments are matched against numerical fields in order to establish the efficacy of the experimental-numerical protocol for obtaining the toughness. We supplement this basic procedure with fractography of the fracture surfaces and ingenuous high speed imaging of the fracture surface in order to gain more insight into the initial process of damage accumulation prior to the initiation of fracture. Our results show that PC does not exhibit an enhancement of fracture tough3

ness with loading rate even at the highest loading rates achievable on a HPB. However, the onset of the initial damage is significantly delayed in case of high rate loading. In fact, the hydrostatic stress level at which damage ensues is significantly higher at high rates compared to a static situation. Further, simulations motivated by ultra high speed imaging and fractography enables us to very clearly delineate the very early stages of damage growth in PC. The damaging process seems to consist of the nucleation of a single micron sized elliptical defect slightly ahead of the notch tip, followed by growth of crazes from the defect surface towards the notch. Moreover, the damaging process is much slower under quasi-static situation than at high rates. While the role of cavitation and crazing towards failure in PC has been discussed in the literature [Agrawal and Pearsall (1991), Hull and Owen (1973), Ishikawa et al. (1977), Legrand (1969), Mills (1976), Narisawa et al. (1980)], the exact timing and sequence of events leading to failure and the quantification of the critical stresses and strains required to trigger them under high rates of loading, to the best of our knowledge, has not been established before. The rest of the paper is organised in the following manner. In the next section, we describe the constitutive model used in the FE simulations. As mentioned above, the calibration of the model for PC has been done under static conditions. We also list the parameters obtained from the calibration exercise. Details of the experimental set-ups, procedures for extracting quantities needed for the numerical simulations and for performing DIC are presented in Sec. 3. The numerical procedure for determining the dynamic J integral through a domain integral procedure is also given here. Results from static and dynamic tests are presented and discussed in Sec. 4. Salient conclusions from this work are presented in Sec. 5. In the following, vectors are denoted by smallcase Roman symbols while Cartesian tensors are denoted by uppercase ones. Tensor scalar products of two tensors A and B are denoted as A : B while the dyadic product is denoted as A ⊗ B. Also, ()0 represents the deviatoric part of a tensor. Further, indicial notation is also used at times and the summation convention is implied unless otherwise mentioned. 2. Constitutive Model When crazing does not take place or is suppressed, as in compression or shear tests, amorphous polymers like PC can undergo quite large strains (up to about 100 %). Their response shows softening upon yielding followed by progressive hardening as the deformation continues. In an analysis of the stress and strain fields around the tip of a blunted crack under mode I, Lai and Giessen (1997) showed that the softening is intrinsic in nature and necessary to capture the localized strain fields observed experimentally as in Ishikawa et al. (1977). We start out with the constitutive description of amorphous polymers at large plastic strains for temperatures below the glass transition Tg . The constitutive model is based on the formulation of Boyce et al. (1988), but we use a

4

modified version introduced by Wu and Giessen (1993). Details of the governing equations and the computational aspects can be found in Wu and Giessen (1996). The reader is also referred to the review by Giessen (1997) together with a presentation of the thermo-mechanical framework in Basu and Giessen (2002). Recently, Kattekola et al. (2014) critically reviewed this constitutive model by considering quasi-static fracture problem for thin PC sheets. The constitutive model makes use of the decomposition of the rate of deformation D into an elastic (De ) and a plastic (Dp ) part as D = De + Dp . Prior to yielding, no plasticity takes place and Dp = 0. In this regime, most amorphous polymers exhibit visco-elastic effects but these are neglected here since we are primarily interested in the effect of the bulk plasticity. Assuming the elastic strains to remain small, the constitutive model takes the form, O

σ = Le : De ,

(1)

O

where σ is the Jaumann rate of the Cauchy stress given in terms of the spin tensor W as O σ = σ˙ − W σ + σW , (2) and Le the usual fourth-order isotropic elastic tensor given by Le = λI2 ⊗ I2 + µI4 .

(3)

Here, λ and µ are Lame’s constants, while I2 and I4 are the symmetric second order and fourth order identity tensors. Assuming that the yield response is isotropic, the isochoric visco-plastic strain rate r 1 0 0 γ˙ p 0 Dp = √ σ ¯, with τ = σ ¯ :σ ¯, (4) 2 2τ p is specified in terms of the equivalent shear strain rate γ˙ p = Dp : Dp , the driving stress σ ¯ = σ − b and the related equivalent shear stress τ . The back stress tensor b describes the progressive hardening of the material as the strain increases and will be defined later. The equivalent shear strain rate γ˙ p is taken from Argon’s (1973) expression " (  5/6 )# −As0 τ γ˙ p = γ˙ 0 exp 1− for T < Tg , (5) T s0 where γ˙ 0 and A are material parameters and T the absolute temperature [note that plastic flow is inherently temperature dependent through (5)]. In Eq. 5 the shear strength s0 is related to elastic molecular properties in Argon’s original formulation but is considered here as a separate material parameter. Further, in order to account for the effect of strain softening and for the pressure dependence of the plastic strain rate, s0 in (5) is replaced by s + αp, where α is a pressure sensitivity coefficient and p = −1/3 trσ. The shear strength s is taken to evolve from the initial value s0 with the plastic strain rate through s˙ = h (1 − s/sss ) γ˙ p , 5

(6)

so as to incorporate strain softening in a simple way. Here, h controls the rate of softening, while sss represents the final, steady state value of s. Completion of the constitutive model requires the description of the progressive hardening of amorphous polymers upon yielding due to deformationinduced stretch of the molecular chains. This effect is incorporated through the back stress b in the driving shear stress τ in Eq. 4. Its description is based on the analogy with the stretching of the cross-linked network in rubber elasticity, but with the cross-links in rubber being replaced with the physical entanglements in an amorphous glassy polymer [Boyce et al. (1988)]. The deformation of the resulting network is assumed to derive from the accumulated plastic stretch [Wu and Giessen (1993)] so that the principal back stress components bα are functions of the principal plastic stretches λβ as X b= bα (epα ⊗ epα ) , bα = bα (λβ ) , (7) α

in which epα are the principal directions of the plastic stretch. In a description of the fully three-dimensional orientation distribution of non-Gaussian molecular chains in a network, Wu and Giessen (1993) showed that b can be estimated accurately with the following combination of the classical three-chain model and the eight-chain description [Arruda and Boyce (1993)]: bα = (1 − ρ) b3−ch + ρb8−ch , (8) α α √ ¯ N is based on the maximum plastic stretch where the fraction ρ = 0.85λ/ ¯ = max (λ1 , λ2 , λ3 ) and on N , the number of segments between entanglements. λ √ The a limit stretch of √ use of Langevin N statistics for calculating bα implies N . The expressions for the principal components of b3−ch and b8−ch contain α α a second material parameter: the initial shear modulus CR = nkB T , in which n is the volume density of entanglements (kB is the Boltzmann constant). For the 3-chain network model the principal back stresses in terms of the accumulated plastic stretches (λα ) are given by the expression [Boyce et al. (1988)]:   λα 1 √ 3−ch −1 √ bα = CR N λα L . (9) 3 N For the 8-chain network model the principal back stresses in terms of the accumulated plastic stretches (λα ) are given by the expression [Arruda and Boyce (1993)]:   1 √ λ2α −1 λc √ b8−ch = C N L , (10) R α 3 λc N with λ2c =

3 1X 2 λβ . 3

(11)

β=1

Using the rate tangent formulation for this constitutive model due to Peirce et al. (1984), Wu and Giessen (1996), the model is implemented using UMAT/ 6

Table 1: Material Parameters for PC

E

ν

2100 0.38

s0

sss

h

A

γ˙ 0

Cr

N

α

99

80

520

300

8.68×1020

15

3

0.08

VUMAT feature in ABAQUS v6.10. To determine the parameters of the constitutive model, uni-axial compression tests are performed on PC cylindrical specimens of diameter 5.3 mm and length 5.3 mm and uni-axial tensile tests on 5.3 mm thick dog bone specimens (dimensions according to ASTM-D638, type I specimen) in the strain-rate range of 10−3 to 1 s−1 /s. During compression tests, end faces of the cylindrical specimens are lubricated with silicone grease and hence little or no barreling of specimens was observed. Elastic properties (Elastic modulus and Poisson’s Ratio) are obtained directly from quasi-static experiments. For other visco-plastic parameters, quasi-static FE simulations of the compressive and tensile tests at corresponding strain rates are performed. Best fit between the engineering stress and engineering strain curves from experiments and from simulations on identical geometries are obtained by varying the parameters using a heuristic approach. While ensuring the best fit, primary attention is devoted to matching the maximum stress before yield, drop in the stress during the softening and rehardening behaviour of the curve. Final set of parameters from the fitting process is listed in Table 1. Only, the pressure sensitivity parameter α, is borrowed from Boyce and Arruda (1990). The reader is also referred to Kattekola et al. (2014) for a more detailed discussion on the fitting procedure. Stress-strain curves from experiments and simulations on identical geometries are compared in Fig. 2, for parameters listed in Table 1. For dynamic simulations density of PC is taken as 1.2 g/cc. 3. Experiments and Numerical Simulations 3.1. Experimental Setups All fracture experiments reported here have been conducted at room temperature on single edge notched specimens in 3-point bend (SENB) loading configuration, as shown in Fig. 3(a), at various rates of loading. In accordance with ASTM standard E1820, the chosen SEN specimen dimensions are 2h/W = 0.5 and 2L/W = 4. The a/W ratio is chosen to be 0.36. Note that the SENB specimen has an almost negligible biaxiality ratio at this value of a/W [Anderson (1994), Fig 3.34]. Specimens are prepared from 5.3 mm thick PC sheets procured locally which are then heat treated (following the protocol suggested by Ishikawa et al. (1977)) in order to remove residual stresses. The fracture toughness of PC has been reported to be very sensitive to the notch tip radius [Fraser and Ward (1977), Inberg and Gaymens (2002), 7

Kattekola et al. (2013)]. Hence, to explicitly identify the effect of loading rate on fracture toughness of PC, controlled notch radius of 150 µm (shown in Fig. 3(b)) has been used in all experiments. The notches have been prepared using a diamond wafering blade. Quasi-static (low loading rate) fracture experiments have been performed on a screw driven universal testing machine (UTM, United Calibration Corp. US, capacity 25 kN) to determine static values of fracture toughness. As discussed earlier, the high loading rate tests have been performed on a HPB setup, as shown in Fig. 4. The setup consists of a loading bar which is 2 m in length and 12.5 mm in diameter. The striker bar is 400 mm in length. Both bars are made of Aluminum-6061 T6 alloy. Ultra high-speed imaging (using SIM02-16, Specialised Imaging Ltd.,UK, max. frame rate - 200 × 106 fps with a 1 MP resolution) is used for all the high strain rate tests reported. The incident (i ) and reflected (r ) strain signals are recorded using two strain gauges mounted diametrically opposite to each other at the mid length of the bar. A set of Ectron 563 strain conditioners having a bandwidth of 200 kHz is used to condition the strain signals, which are subsequently recorded using a NI-PCI 6115 data acquisition card at a sampling rate of 1 MHz. The high-speed camera is triggered using a make trigger circuit placed at the impact end of the bar. After an appropriate initial delay time a set of 16 images at a fixed framing rate are captured during the deformation and fracture of specimen. The Xenon flash lamps used for illumination and the data acquisition system are slave triggered by the high-speed camera, thereby providing time synchronization between the strain signals and the images recorded. In some of the experiments (discussed later) the notch tip is closely observed at an oblique angle in order to record the fracture surface during the loading of the specimen. 3.2. Load-point displacement and fracture initiation time calculation Using one-dimensional wave theory, the displacement history u0 (t) at the end of the bar which is in contact with the specimen is calculated from recorded incident and reflected strains as, Z t u0 (t) = C0 (−i + r ) dt, (12) 0

where C0 is the longitudinal wave velocity in an Aluminum bar (calculated to be 5000 m/s). Synchronization of high-speed images with HPB data provide the crack initiation time within the uncertainty of camera interframe time, which is 1-10 µs (depending on the framing rate used). A typical displacement history with the points corresponding to images just before (t1 ) and after (t2 ) initiation of fracture is shown in Fig. 5. Fracture initiation time (tf ) is then assumed to be the average of t1 and t2 with a possible maximum error of ±(t2 − t1 )/2. 3.3. Digital image correlation (DIC) Measurement of the full-field displacements near notch-tip in selected dynamic experiments has been performed using the DIC technique. To facilitate 8

the use of DIC, a random speckle intensity pattern is applied on the specimen using white and black spray paints, as shown in Fig. 3(a). Two Xenon flash lamps have been used to illuminate the specimen from the front. Typically, an area of dimension 8 mm × 11 mm near the notch is imaged. Each image in the set of 16 taken by the camera is correlated with the corresponding image of the undeformed sample (from the set of 16) taken before test. The correlation is performed using the VIC-2D software (Correlated Solutions, Raliegh, NC, USA). As an example, Fig. 6 shows displacement contours just before fracture initiation from a typical correlation. Fig. 6(a) shows the component of the displacement in the x direction (the bending displacements u(x, y)), while (b) shows the y component (the opening displacements v(x, y)) in the region around the notch tip. These displacements are measured for image taken at a time, just before fracture initiates (see, Fig. 5). These figures demonstrate the fact that DIC along with high speed imaging works in a robust manner in the case of PC and provides reliable information on the full field displacements. Moreover, for an elastic material, the crack-tip opening displacement field under pure mode-I static loading (Williams (1957)) is given as: vth =

n ∞  n  i h X n n n r2 an κ + + (−1)n cos θ − cos −2 θ , 2G 2 2 2 2 n=1

(13)

where r and θ are radial coordinates, G is the shear modulus and κ = 3 − 4ν for plane strain. Out of the constants an , the first one is directly related to the mode-I SIF KI . Equation 13 is fitted to the dynamic opening displacement field (like in Fig. 6(b)) obtained from DIC in a least square sense. Fig. 7 shows the fit obtained by considering the first 7 terms of Eq. 13. Excellent agreement between both the contours not only reiterates the robustness of the DIC procedure at high strain rates, but also confirms the establishment of static crack-tip fields and small-scale yielding conditions at the notch tip before initiation of fracture. 3.4. Finite element (FE) model and its validation Three dimensional dynamic FE simulations have been performed using the commercial FE software ABAQUS v6.10. For all cases reported, eight noded isoparametric hexahedral elements with reduced integration (C3D8R) have been used alongwith explicit time integration in case of dynamic and standard implicit for static cases. A typical FE mesh used in the simulations is shown in Fig. 8. Symmetry boundary conditions are used in the y and z directions so that: w(X, Y, 0) = 0,

(14)

v(X, 0, Z) = 0 on 0 < X < W − a.

(15)

and Thus, only a quarter of the specimen needs to be modelled. The radius of the notch tip is denoted by rt . The entire loading bar is not modelled in simulation. End of the bar and the supports are modelled as frictionless rigid surfaces (with the geometry closely adhering to that of the experimental situation) with surface 9

to surface contact implemented between the surfaces and the specimen, to take care of contact loss during observed in experiments. The point O in Fig. 8 has u = −u0 (t),

(16)

where u0 (t) is the displacement history at the end of the bar obtained from experiments using Eq. 12. This displacement history corresponds to the bar end point velocity of V0 . The fine mesh used close to the notch tip is also shown in Fig. 8. A typical simulation contains 162002 elements and 516975 degrees of freedom. In order to validate the FE scheme, specimen bending displacement uA has been measured using DIC at regions very close to the loading point (at point A in Fig. 8). This displacement is compared with the bar end displacement u0 (t) calculated from Eq. 12 in Fig. 9. In the initial period (up to 60 µs) there is close agreement (within 2%) between the load point displacements calculated by DIC and Eq. 12. However, after this time the displacement calculated using DIC exceeds u0 (t) indicating loss of contact between the specimen and the bar. In fact, loss of contact occurs more than once for the experiment as shown in Fig. 9. By separately imaging the region near the point of impact, we verified that the instances where the displacement from DIC and u0 (t) differ indeed correspond to loss of contact between the specimen and the loading bar. The situation was replicated in a FE simulation where, as discussed already, u0 (t) was applied on the loading bar (point O in Fig. 8). Displacement of the point A in Fig. 8 is further compared to the displacement obtained from DIC in Fig. 9. Clearly, the displacement follows the actual data obtained from DIC very closely and the loss of contact at the loading point is captured faithfully. Crack mouth opening displacement (CMOD), defined as δ = v(−a, rt , Z) − v(−a, −rt , Z), has also been monitored using DIC. The variation of the CMOD with time obtained from DIC and simulation is also shown in Fig. 9. Clearly, they are in excellent agreement (within ±10%). It can be observed that the decrease in CMOD associated with loss of contact is also captured faithfully by the simulation. Figures 7 and 9 demonstrate that the combined experimental-numerical scheme for modelling impact loading of PC using HPB works well with the constitutive model and boundary conditions used. It should be noted that the comparisons between high speed DIC results and simulations presented so far are for far field displacements. We have shown in a recent work [Kattekola et al. (2013)] that as plasticity in PC is confined very close to the notch, even a rudimentary constitutive description can predict the overall far field quantities quite well. Hence, the above comparisons do not test all features of the constitutive model. However, at least for static situations, we have rigorously shown [Kattekola et al. (2013)] that the constitutive description used here, with parameters determined from uniaxial tests at multiple strain rates, does describe the plastic zone ahead of the notch tip well. The dynamic J integral has been calculated using the domain integral method [Shih et al. (1986)] on a contour that collapses on the notch tip. ABAQUS does 10

not provide J integral as direct output in explicit calculations, hence it is calculated using a separately written Python program which uses the domain integral method. For the elasto-viscoplastic material model used, the J integral values are domain independent (within 2%) for domains having more than 100 elements. All J integral values reported in the following sections are measured at ˙ The Z=0. Unless otherwise mentioned, loading rates are defined in terms of J. method for calculating J˙ is explained later. 4. Results and Discussion 4.1. Static fracture The sequence of events leading to fracture in PC have been studied in detail by Ishikawa et al. (1977) under quasi-static situations and by Mills (1976) under dynamic ones. Detailed fractographic studies of the fracture surfaces have been carried out by Hull and Owen (1973) and Agrawal and Pearsall (1991). Sufficiently thick PC specimens with sharp notches invariably exhibit crazing before brittle fracture initiates. Signatures of the initial craze nucleus is seen in fractographic studies. These crazes appear some distance ahead of the notch tip, almost at the midplane and with the fibrils oriented in the direction of the maximum principal stretch. Estimates of hydrostatic stress required for voiding followed by crazing have also been determined by a range of combined numerical-experimental techniques [Ishikawa et al. (1977), Kambour and Farraye (1984), Kambour et al. (1986), Nimmer and Woods (1992)]. These studies indicate that a hydrostatic stress of 80 − 90 MPa is necessary to initiate voids that lead to crazing. In this section, we will show that under quasi-static conditions, our experiments reproduce all the features observed by earlier authors quite well. In the quasi-static experiments, the SENB specimen is loaded at a cross-head speed of 0.5 mm/min. During the test, the notch tip is observed closely at an oblique view, using a 2 MP CCD camera at a frame rate of 5 fps. A sequence of events at the notch tip is shown in Figs. 10(a)-(f). Similar to Ishikawa et al. (1977) and Mills (1976), we too observe that fracture initiates with the appearance of one or more isolated defects some distance ahead of the notch tip (Figs. 10(b)-(d)). In fact, the defect nearest to the notch is at a distance of 300 µm ahead of it. At the stage when the defect is visible, its diameter is around 100 µm. With loading, these defects are seen to expand and coalesce forming a semi-elliptical process zone spanning the specimen thickness in front of the notch (Figs. 10(e),(f)). The semi-elliptical front now progresses rapidly in its plane and the sample fractures in a brittle manner. The semi elliptical zone is also clearly seen in fractographs taken in an SEM (see Fig. 11). Extent of the zone is marked with a dotted line. The morphology of this zone conforms to the description by Hull and Owen (1973). Two initial defect locations are visible (marked A and B in Fig. 11). The one on the left is rough indicating, according to Hull and Owen (1973), that crazes on several planes were generated. Finally, propagation occurred along one of these

11

crazes. The region on the right is smooth indicating that propagation occurred along the craze-bulk interface. In most fractographs for samples fractured under static conditions, two regions of initiation are visible. The initial defects appear as islands surrounded by a relatively smoother region, and grow in an almost radially symmetric manner by craze breakdown. The smooth regions indicate propagation along craze-bulk interface. The smoother region outside the ellipse is marked by narrow radial protrusions which, according to Hull and Owen (1973) are marks of tearing due to propagation either along the midplane of the craze or show that the fracture path keeps alternating between the two craze-matrix boundaries as it grows in the radial direction. The above sequence of events allows us to compute the mean stress required for defect nucleation in PC. We know that the location of first defect is ∼ 300 µm from the notch tip almost at the mid-plane of the specimen. Loaddisplacement curve for the experiment is shown in Fig. 12(a). Circles on the curve correspond to the images shown in Fig. 10. Clearly, the defect nucleates early in the deformation at J = 3.5 kN/m while brittle crack propagation starts at J = 13 kN/m. Computationally obtained variation of the hydrostatic stress σm = σkk /3, with distance ahead of the notch tip at J = 3.5 kN/m is shown in Fig. 12(b). The variation of the equivalent plastic strain γp with distance ahead of the notch is also shown in the same figure. At a distance 300 µm ahead of the notch tip (where, the first defect is seen to appear in the experiments), the hydrostatic stress attains a peak value of 80 MPa. Also, at this location, the value of the equivalent plastic strain is low indicating that a purely mean stress based criterion for defect initiation in PC is essentially correct. Therefore we conclude that, under static conditions, the value of the critical hydrostatic stress required to nucleate a defect is 80 MPa (See Fig. 12(b)). The above analysis gives a quantitative value of the critical hydrostatic stress required to initiate a visible defect. However, under static situations the J at fracture initiation is almost 4 times that at which the defect becomes visible. Therefore, the quantitative criterion for the initiation of fracture still eludes us. We will return to this question after discussing the results from the dynamic tests and simulations. 4.2. Dynamic fracture The sequence of events leading to the initiation of dynamic fracture is similar to the static case. Similar to the static case, the region near the notch tip has been observed with the high speed camera in an oblique view. Figs. 13(a) through (e) show a sequence of photographs taken at intervals of 1 µs. These sequence of images should be seen in conjunction with the evolution of J with time shown in Fig. 14 (for a case with V0 = 12 m/s). No visible defects nucleate before 80 µs, at which time J = 11 kN/m. Fracture ensues immediately thereafter at 84 µs or J = 12 kN/m. The basic nature of defect growth is quite identical to the static case. Between 80 to 84 µs, the initial defect coalesces with others to form a flat semielliptical process zone. Unlike the static case however, this process zone does not span the thickness till at the point of fracture initiation. 12

The fractograph of a dynamically fractured surface shown in Fig. 15, is not very different from the static one . The region where the initial defect has nucleated is again clearly visible and is marked with a dotted line. Note that the defects with rough surfaces that were present in the static case are not seen here. The first defect shown in Fig. 15 is absolutely smooth indicating that, unlike the static case, crazing in multiple planes did not happen in this case. A single defect nucleated and led to failure. The subsequent process involving the expansion of this defect seems to be identical to the static situation. The initial defect propagates in a radially symmetric manner. The circumferential crack front propagates by alternating between the top and the bottom craze-bulk interfaces as discussed for the static case. The apparent similarity between the static and dynamic situations in the nature of the initial defect as well as the manner in which the process zone propagates, suggests that a single initial defect is responsible for failure in the dynamic case while several initial defects in parallel planes are formed in the static case before fracture propagates along one of them. Subsequently, the mechanism of propagation seems to be the same in the two cases. In other words, at high strain rates, there does not seem to be a drastic switch in the failure mechanism. The only major difference lies in the fact that multiple defect nucleations consume a lot of energy in order to cause fracture initiation in static case while the initiation toughness of the single defect and subsequent rapid fracture are almost equal in the dynamic case. A few words about the nature of variation of J integral with time is in order here. The load on the specimen (which is the contact force between the loading bar and the specimen) as a function of time is shown in Fig. 16(a) for V0 = 6, 10 and 12 m/s. Note that the time taken for a longitudinal wave to travel from the loading point to the notch tip is τref ∼ 5 µs. The load has been plotted till the time to fracture initiation noted from the corresponding experiments. The corresponding variations in J with time are shown in Fig. 16(b). The specimens, irrespective of the impact velocity, are loaded monotonically upto 25 µs and the load drops thereafter, leading to loss of contact between the loading bar and the specimen at 60 µs. Corresponding to the load drop, the evolution of the J integral slows down. Note also that the drop in or flattening of the J integral versus t plot also manifests as a drop in the CMOD as shown in Fig. 9. All the three experiments involve multiple impacts on the specimen. For the highest velocity of impact, the specimen fails before the second impact can take place. The other two cases with slower impact velocities fail after two and three impact events. The calculation of an effective J˙ in cases where unloading and loss of contact occurs is complicated by the fact that each loss of contact ˙ We have taken event leads to a drop in J while re-contact leads to a higher J. ˙ the effective J to be the slope of the J-time plot at the instant where fracture initiates. Recall that this time is determined from experiments. The variation of the hydrostatic stress and equivalent plastic strain with distance ahead of the notch tip at 81 µs (at the instant the first defect nucleates) is shown in Fig. 17(a). The hydrostatic stress required for defect initiation is now 115 MPa, significantly higher than the static case. The fact that the 13

hydrostatic stress required for defect initiation increases with rate has been shown also by Narisawa et al. (1980), albeit at much lower rates. Further, the variation of the critical hydrostatic stress necessary for defect initiation with effective loading rate J˙eff has been shown in Fig. 17(b). To the best of our knowledge, the variation of the critical hydrostatic stress required to initiate a defect ahead of a notch in PC over such a large range of loading rates has not been quantified before. Since the times to defect as well as fracture initiation (i.e tdef and tfrac respectively) are known from experiments, the critical values of J, Jdef and Jfrac can easily be extracted from simulations. These variations with J˙eff have also been plotted in Fig. 18. Note that the defect initiates at tdef = 81 µs in the case shown in Fig. 14 and fracture initiates at tfrac = 84 µs, just 3 µs later. In fact, in all cases loaded at high impact velocities i.e. with J˙eff > 1 × 104 kN/m − s, the interval between tdef and tfrac is very short. Fracture initiates immediately after a defect does and Jdef ' Jfrac . This is not the case under static situation where, Jfrac >> Jdef , as shown earlier in Fig. 12. Moreover, while the toughness at defect initiation Jdef is significantly lower in the static case, the toughness at fracture initiation Jfrac shows only a very weak dependence on loading rate. These observations have to be reconciled with the fact that there are suitable differences between the fracture surfaces observed for static and dynamic situations, specially in the shapes before rapid crack propagation. Therefore, is there a universal criterion which needs to be satisfied for fracture to initiate and is valid across the range of loading rates? The above question can only be addressed through simulations. To this end, we performed simulations with a defect (in the form of a spherical void) centered at the location of highest hydrostatic stress obtained ahead of the notch in simulations without voids (see Figs. 12(b) and 17(a)). The size of the void is comparable to the first defect seen in static and dynamic cases, i.e. dv = 100 µm and is located on the mid-plane 300 µm away from the notch tip. Obviously, by modelling the existence of a defect from the very onset, we are ignoring the fact that a critical hydrostatic stress needs to build up to initiate it. However, as shown in Figs. 12(b) and 17(a), the plastic strain at the location of the void is very low. Initiated under elastic conditions, the defect is likely to have nucleated not too long before tdef , when it attains a visible size. Additionally, the visible initial defect in PC may be a craze rather than a void. However, our motivation in this part of the work is to postulate a suitable criterion for fracture initiation. To this end, what matters is the interaction between the defect and the notch. The growth dynamics of the defect (which will obviously be different for a craze and a void) is not important here. The geometry that is simulated is shown in the inset of Fig. 19(a). The variation of the plastic stretch normal to the plane of the notch (X − Z plane), λpy along X has been shown in Figs. 20(a) and (b) for the same values of J. This quantity is of interest since experiments indicate [Estevez and Giessen (2005), Gearing and Anand (2004), Kramer (1983)] that a critical value of plastic stretch is necessary for craze failure. In both Figs. 20(a) and (b) the solid curve 14

corresponds to the instant where fracture initiates in the experiments. Firstly, note from the contours in Fig. 20(a) that λpy is high around the void periphery in the static case. In the dynamic case too, λpy is higher in the part of the periphery facing the notch. These contours indicate the possible directions in which the crazes emanating from the void will propagate. Indeed, these profiles are consistent with radial craze propagation indicated in Figs. 11 and 15. More importantly, at the instant of fracture initiation, (J = 12 kN/m in Figs. 20(a) and (b)) the maximum values of λpy is attained at the void periphery and notch tip in the static case and at the void periphery in the dynamic. Remarkably, in both cases maximum values of λpy is about 1.8 at the initiation of fracture. It has been suggested by Gearing √ and Anand (2004) that a critical values of effective plastic stretch λe = √13 trB p is necessary to cause fracture in polymers. For PC, they suggested a critical values of 1.192 for λe . Our observation that attainment of λpy = 1.8 causes fracture initiation in PC is in line with Gearing and Anand (2004). Moreover, in the dynamic case, the existence of the defect seems essential for attainment of the critical value of λpy , as the value of the plastic stretch is low every where else except at the defect periphery. 5. Conclusion The events leading to fracture in PC over a very wide range of loading rates ˙ has been studied. (quantified by the rate of increase of the J-integral, J) ˙ The values of J at which a defect initiates seems to depend strongly on J. Defect initiation occurs by a purely mean stress dependent mechanism, without appreciable plastic deformation at 80 MPa in static and 115 ± 5 MPa at J˙ = 104 to 106 kN/m − s. The mean stress required for defect initiation is an increasing function of J˙ and has been quantified for a wide range of J˙ values. The defect initiation toughness is low in the static case (Jdef = 3.5 kN/m) compared to dynamic (Jdef ≈ 12 kN/m at J˙ = 104 − 106 kN/m − s). ˙ However, J at the initiation of fracture does not seem to depend on J, and Jfrac is around 12 ± 3 kN/m across a very wide range of loading rates. In the static case Jfrac >> Jdef because a large number of defects nucleate on parallel planes ahead of the notch tip and grow before fracture occurs. In highly dynamic cases Jfrac ≈ Jdef and a single defect nucleate and leads to fracture almost instantly. Experiments with high speed imaging and supplementary numerical simulations have allowed us to identify two important critical conditions quantitatively. Finally, the critical mean stress required for defect initiation ranges from 80 MPa under static conditions to 115 ± 5 MPa at J˙ = 104 to 106 kN/m − s. ˙ when Secondly, fracture initiation happens, irrespective of the loading rate J, p the normal plastic stretch λy = 1.78 ± 0.2. Identification and quantification of these criteria will help to increase the predictive abilities of simulations on PC.

15

Acknowledgments The financial support through grant number SR/FST/ETII-003/2006 under the FIST program by Department of Science and Technology, Government of India for the Ultra-high speed camera used in this study is acknowledged. References Agrawal, C. M., Pearsall, G. W., 1991. The fracture morphology of fast unstable fracture in polycarbonate. Journal of Materials Science 26 (7), 1919–1930. Allen, G., Morley, D. C. W., Williams, T., 1973. The impact strength of polycarbonate. Journal of Material Science 8 (10), 14491452. Anderson, T. L., 1994. Fracture Mechanics: Fundamentals and Applications, Second Edition. CRC Press. Arruda, E. M., Boyce, M. C., 1993. A three-dimensional constitutive model for the large stretch behaviour of rubber elastic materials. Journal of Mechanics and Physics of Solids 41 (2), 389–412. Basu, S., Giessen, E. V., 2002. A thermo-mechanical study of mode i, small-scale yielding crack-tip fields in glassy polymers. International Journal of Plasticity 18 (10), 1395–1423. Belenky, A., Bar-On, I., Rittel, D., 2010. Static and dynamic fracture of transparent nanograined alumina. Journal of the Mechanics and Physics of Solids 58 (4), 484–501. Boyce, M. C., Arruda, E. M., 1990. An experimental and anaiytical investigation of the large strain compressive and tensile response of glassy polymers. Polymer Engineering and Science 30 (20), 1288–1298. Boyce, M. C., Parks, D. M., Argon, A. S., 1988. Large inelastic deformation of glassy polymers. part i: Rate dependent constitutive model. Mechanics of Materials 7 (1), 15–33. Casiraghi, T., 1978. The fracture mechanics of polymers at high rates. Polymer Engineering and Science 18 (10), 833839. Chichili, D. R., Ramesh, K. T., 1995. Dynamic failure mechanisms in a 6061-t6 al/al2o3 metalmatrix composite. International Journal of Solids and Structures 32, 2609–2626. Estevez, R., Basu, S., 2008. On the importance of thermo-elastic cooling in the fracture of glassy polymers at high rates. International Journal of Solids and Structures 45, 34493465. Estevez, R., Giessen, E. V., 2005. Modeling and computational analysis of fracture of glassy polymers. Advances in Polymer Sciences 188, 195234. 16

Fraser, R. A. W., Ward, I. M., 1977. The impact fracture behaviour of notched specimens of polycarbonate. Journal of Materials Science 12 (3), 459–468. Gearing, B. P., Anand, L., 2004. Notch-sensitive fracture of polycarbonate. International Journal of Solids and Structures 41 (3-4), 827845. Giessen, E. V., 1997. Localized plastic deformations in glassy polymers. European Journal of Mechanics - A/Solids 16, 87–106. Huang, S., Luo, S., Xia, K. (Eds.), Jun. 2009. Dynamic fracture initiation toughness and propagation toughness of PMMA. Albuquerque, New Mexico, USA. Hull, D., Owen, T. W., 1973. Interpretation of fracture surface features in polycarbonate. Journal of Polymer Science: Polymer Physics Edition 11 (10), 2039–2055. Inberg, J. P. F., Gaymens, R. J., 2002. Polycarbonate and co-continuous polycarbonate/abs blends: influence of notch radius. Polymers 40 (15), 4197–4205. Ishikawa, M., Narisawa, I., Ogawa, H., 1977. Criterion for craze nucleation in polycarbonate. Journal of Polymer Science: Polymer Physics Edition 15 (10), 1791–1804. Kambour, R. P., Farraye, E. A., 1984. Crazing beneath notches in ductile glassy polymers: a materials correlation. Polymer Communications 25 (12), 357–360. Kambour, R. P., Vallance, M. A., Farraye, E. A., Grimaldi, L. A., 1986. Heterogeneous nucleation of crazes below notches in glassy polymers. Journal of Material Science 21 (7), 2435–2440. Kattekola, B., Desai, C., Parameswaran, V., Basu, S., 2014. Critical evaluation of a constitutive model for glassy polycarbonate. Experimental Mechanics 54 (3), 357368. Kattekola, B., Ranjan, A., Basu, S., 2013. Three dimensional finite element investigations into the effects of thickness and notch radius on the fracture toughness of polycarbonate. International Journal of Fracture 181 (1), 112. Koyabashi, T., Matsunuma, K., Ikawa, H., Motoyoshi, K., 1988. Evaluation of static and dynamic fracture toughness in ceramics. Engineering Fracture Mechanics 31 (5), 873–885. Kramer, E. J., 1983. Microscopic and molecular fundamentals of crazing. In: Kausch, H. H. (Ed.), Crazing in Polymers. Vol. 52-53 of Advances in Polymer Science. Springer Berlin Heidelberg, pp. 1–56. Lai, J., Giessen, E. V., 1997. A numerical study of crack-tip plasticity in glassy polymers. Mechanics of Materials 25 (3), 183–197.

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Legrand, D. G., 1969. Crazing, yielding, and fracture of polymers. i. ductile brittle transition in polycarbonate. Journal of Applied Polymer Science 13 (10), 21292147. Mills, N. J., 1976. The mechanism of brittle fracture in notched impact tests on polycarbonate. Journal of Material Science 11 (2), 363375. Narisawa, I., Ishikawa, M., Ogawa, H., 1980. Notch brittleness of ductile glassy polymers under plane strain. Journal of Material Science 15 (8), 2059–2065. Nimmer, R. P., Woods, J. T., 1992. An investigation of brittle failure in ductile, notch-sensitive thermoplastics. Polymer Engineering and Science 32 (16), 1126–1137. Peirce, D., Shih, C. F., Needleman, A., 1984. A tangent modulus method for rate dependent solids. Computers and Structures 18 (5), 875887. Ravetti, R., Gerberich, W. W., Hutchinson, T. E., 1975. Toughness, fracture markings, and losses in bisphenoi-a polycarbonate at high strain-rate. Journal of Material Science 10 (8), 14411448. Ravi-Chander, K., Knauss, W. G., 1984. An experimental investigation into dynamic fracture: I. crack initiation and arrest. International Journal of Fracture 25 (4), 247–262. Ravi-Chander, K., Yang, B., 1997. On the role of microcracks in the dynamic fracture of brittle materials. Journal of the Mechanics and Physics of Solids 45 (4), 535–563. Rittel, D., 1998. Experimental investigation of transient thermoelastic effects in dynamic fracture. International Journal of Solids and Structures 35 (22), 2959–2973. Rittel, D., Maigre, H., 1996. An investigation of dynamic crack initiation in pmma. Mechanics of Materials 23 (3), 229–239. Rittel, D., Rosakis, A. J., 2005. Dynamic fracture of berylium-bearing bulk metallic glass systems: A cross-technique comparison. Engineering Fracture Mechanics 72 (12), 1905–1919. Shah, Q. H., 2009. Impact resistance of a rectangular polycarbonate armor plate subjected to single and multiple impacts. International Journal of Impact Engineering 36 (9), 1128–1135. Shih, C. F., Moran, B., Nakamura, T., 1986. Energy release rate along a threedimensional crack front in a thermally stressed body. International Journal of Fracture 30 (2), 79–102. Wada, H., 1992. Determination of dynamic fracture toughness for pmma. Engineering Fracture Mechanics 41 (6), 821–831. 18

Wada, H., M.Seika, Calder, C. A., 1993. Measurement of impact fracture toughness for pmma with single-point bending test using an air gun. Engineering Fracture Mechanics 46 (4), 715–719. Weerasooriya, T., Moy, P., Casem, D., Cheng, M., Chen, W., 2006. A four-point bend technique to determine dynamic fracture toughness of ceramics. Journal of the American Ceramic Society 89 (3), 990–995. Williams, M. L., 1957. On the stress distribution at the base of a stationary crack. Journal of Applied Mechanics 24, 109–114. Wright, S. C., Fleck, N. A., Stronge, W. J., 1993. Ballistic impact of polycarbonate-an experimental investigation. International Journal of Impact Engineering 13 (1), 120. Wu, P. D., Giessen, E. V., 1993. On the improved network model for rubber elasticity and their applications to orientation hardening in glassy polymers. Journal of Mechanics and Physics of Solids 41 (3), 427–456. Wu, P. D., Giessen, E. V., 1996. Computational aspects of localized deformations in amorphous glassy polymers. European Journal of Mechanics, A/Solids 15 (5), 799–823. Yokoyama, T., 1995. Determination of dynamic fracture-initiation toughness using a novel impact bend test procedure. Journal of Pressure Vessel Technology 155 (4), 389–397. Zhou, J., Wang, Y., Xia, Y., 2006. Mode-i fracture toughness of pmma at high loading rate. Journal of Material Science 41 (24), 8363–8366.

19

14 Wada (1992) 12

Wada et al. (1993) Rittel et al. (1996)

√ KI d (M P a m)

10 8

Zhou et al. (2006) Huang et al. (2009)

6 4 2 0 −3 −2 −1 0 1 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 √ K˙ I d(M P a m/s) Figure 1: Effect of loading rate on fracture toughness of PMMA.

20

140 Expreiment 120

σ (MPa)

100

Simulation 1 s−1 (Compression)

80 0.001 s−1 (Compression)

60 40

0.001 s−1 (Tension)

20 0 0

0.1

0.2

²

0.3

0.4

Figure 2: Uni-axial tension and compression curves for PC.

21

0.5

22

Z

(b)

Figure 3: (a) The SEN specimen geometry with speckle pattern in the region on which DIC is performed and (b) A close up of the semi-circular notch of 150 µm radius made

(a)

X

Y

Flash Lamp Striker Bar

Loading Bar Rigid Support

Make Trigger Trigger In

Trigger In

Trigger Out

Camera

SG Amplifier

Figure 4: Schematic of dynamic experiment test setup

23

1 0.9 0.8 0.7

After fracture

0.6 0.5 0.4 Before fracture

0.3 0.2 0.1 0 0

20

40

60

80

100

Figure 5: Typical bar end displacement history obtained using Eq. 12, and corresponding images before and after fracture, from a dynamic experiment

24

u (mm) 0.9 3 2

0.85

Y (mm)

1 0

Notch

0.8

−1 0.75

−2 −3 −2

0

2 X (mm)

4

6

0.7

(a)

v (mm) 0.25 3 0.15

2

Y (mm)

1 0

0.05 Notch −0.05

−1 −2

−0.15 −3 −2

0

2 X (mm)

4

6

−0.25

(b) Figure 6: (a) Bending (u) displacement, and (b) opening (v) displacement contours from DIC

25

0.2

0.15

4 0.25 0.2 0.15 3

0.1 0.1

0.05 0.05

0

0 DIC

-0.05 Simulation -0.05 Static elastic

0

crack-tip field

2

−1

0.15 0.1 0.05 0.05 0.2 0.15 0 0.1 0.1 Notch 0 -0.05 -0.10.05 0.1 0.2 0.15 0.05

−2

-0.2 -0.15 -0.1 -0.05

Y (mm)

1 0

0.2

0 0 00 0 0

00 0

0

0 0.05

0

−3 -0.15 -0.25 -0.2 −4 0.2 0.15 −2

0

0

0.1 -0.1 0

-0.05 0.05 2 X (mm)

0.05

0 0 4

-0.05

6

Figure 7: Comparison of full-field opening (v) displacements (in mm) obtained from DIC, Simulation and Eq. 13

26

RP

B

RP

O

Y

A

X

Z C

Figure 8: Three dimensional FE model for simulation

27

1.2 u0 (t)Eq.12 1

uA (t)DIC

u0 , uA , δ (mm)

uA (t)Simulation 0.8

δDIC δSimulation

0.6

Contact Loss

0.4

0.2

0 0

30

60

90 t (µs)

120

Figure 9: Displacement and CMOD comparison

28

150

180

29 (e) Load=273 N

(b) Load=206 N

(f) Load=368 N

(c) Load=221 N

Figure 10: Oblique view of notch tip under quasi-static loading, showing static fracture process. The undeformed notch is shown in (a). The first defect (circled) initiates at a distance of 300 µm at a load level of 206 N (b). The evolution of the damaged zone with increasing load is shown in (b) through (e).

(d) Load=228 N

(a) Load=0 N

30

Initial Notch

300 μm

20 μm

Figure 11: SEM micrograph of the fractured surface of the specimen shown in Fig. 10. The points A and B indicate the approximate locations of the initial defects

500 μm

Defect intiation zone

400 J =13.2 kN/m 350

f

Load (N)

300

e

250 b

200

cd

150 J =3.5 kN/m 100 50 0 0

a 0.5 1 Displacement (mm)

1.5

120

0.3

100

0.25

80

0.2

60

0.15

40

0.1

20

0.05

0 0

γp

σm (MPa)

(a)

0.2

0.4 0.6 x (mm)

0.8

0 1

(b) Figure 12: (a) Load-Displacement curve for a typical quasi-static fracture test. The circles correspond to the stages shown in Fig. 10. (b) Variation of hydrostatic stress and equivalent plastic strain ahead of the notch tip under static loading at J = 3.5 kN/m when the first 31 defect appears.

32 (d) t=83µs

(b) t=81µs

(e) t=84µs

(c) t=82µs

Figure 13: High speed images taken at intervals of 1 µs, (a) Just before defect initiation, (b) at the instant when the first defect nucleates and for three microseconds thereafter (c)-(e).

(a) t=80µs

14 Before final fracture

12

J (kN/m)

10 Defect initiation

8 6

Undeformed

Notch tip

4

Notch

2 0 0

10

20

30

40

50

60

70

Figure 14: J integral history for dynamic fracture test

33

80

90

34

500 μm

Initial Notch

Figure 15: SEM micrograph of defect initiation zone in dynamic case

200 μm

Defect intiation zone

1400 V0 =6 m/s 1200

V0 =10 m/s V0 =12 m/s

Load (N)

1000 800 600 400 200 0 0

50

100 t (µs)

150

(a)

20 V0 =6 m/s V0 =10 m/s

J (kN/m)

15

V0 =12 m/s

10

5

0 0

50

100 t (µs)

150

(b) Figure 16: (a) Contact force and between the loading bar and the specimen. (b) The variation of the J Integral with time for V0 = 6, 10 and 12 m/s.

35

120

0.4

90

0.3

60

0.2

30

0.1

γp

0.5

σm (MPa)

150

0 0

0.2

0.4 0.6 x (mm)

0 1

0.8

(a)

130 120

def σm (MPa)

110 100 90 80 70 60 −1 10

0

10

1

10

2

3

4

10 10 10 J˙eff (kN/m-s)

5

10

6

10

7

10

(b) Figure 17: (a) Variation of hydrostatic stress and equivalent plastic strain ahead of the notch tip under dynamic loading, and (b) variation of critical mean stress required for void initiation with J˙eff

36

18 dynamic dynamic Jdef and Jfrac

16 14

J (kN/m)

12 10 8 static Jfrac

6 4 2 0 −2 10

static Jdef

−1

10

0

10

1

10

2

10 J˙ef f

3

4

10 10 (kN/m-s)

5

10

6

10

Figure 18: The value of the J Integral at the instant of initiation of a visible defect Jdef and ˙ The rate J˙ is calculated from the and fracture Jfrac plotted as a function of loading rate J. slope of the J verses time curve at the instant fracture initiates.

37

7

10

38

0

0

Void Z

Y

0.1

Notch tip

X

(a)

0.3

X (mm)

0.2

0.4

-0.2

0

1.7 1.6 1.5 0 1.4 1.3 1.2 1.1 -0.1 1

0.1 1.8

1.9

0.2

0.1

(b)

0.3

X (mm)

0.2

0.4

1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1

Figure 19: Contours of plastic stretch λpy plotted at Jfrac for the (a) static and (b) dynamic (V0 = 12 m/s) cases. The contours are plotted on the X − Z plane.

-0.2

-0.1

Z (mm)

0.1

0.2

Z (mm)

39 (a)

1 0

0.6

1 0 0.5

1.1

1.1 0.4

1.2

1.2

0.3 x (mm)

1.3

1.5

1.6

1.7

1.8

1.3

0.2

J =4 kN/m

J =8 kN/m

1.4

0.1

Void

1.9

2

1.4

1.5

1.6

1.7

1.8

1.9

J =12 kN/m

0.1

0.2

(b)

0.3 x (mm)

Void

0.4

0.5

J =4 kN/m

J =8 kN/m

J =12 kN/m

0.6

Figure 20: Variation of the opening plastic stretch λpy on Z = 0 ahead of the notch tip for (a) the static and (b) dynamic (V0 = 12 m/s) cases. The solid curve at J = 12 kN/m correspond to the instant at which fracture initiates.

λyp

2

λyp