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Cohesive zone laws for void growth – I. Experimental field-projection of crack-tip crazing in glassy polymers

Soonsung Hong1, Huck Beng Chew2 and Kyung-Suk Kim2, *

1. Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA

2. Division of Engineering, Brown University, Providence, RI 02912, USA

Abstract

A hybrid framework for inverse analysis of crack-tip cohesive zone model isdeveloped in this two-part paper to measure cohesive zone laws of void growth inpolymers by combining analytical, experimental and numerical approaches. This paper focuses on experimental measurements of the cohesive zone laws for two nonlinear fracture processes in glassy polymers, namely multiple crazing in crack-growth toughening of rubber-toughened HIPS and crazing of steady-state crack growth in PMMA under a methanol environment. To this end, electronic speckle pattern interferometry (ESPI) is first applied to measure the crack-tip displacement fields surrounding the fracture process zones in these polymers. These fields are subsequentlyequilibrium-smoothed and used in the extraction of the cohesive zone laws via an analytical solution method of the inverse problem, the planar field projection method (P-FPM) (Hong and Kim, 2003). Results show that the proposed framework of the P-FPM could provide a systematic way of finding the shape of the cohesive zone laws governed by the different micro-mechanisms in the fracture processes. In HIPS, inter-particle multiple crazing develops and the craze zone broadens ahead of a crack-tip under mechanical loading. The corresponding cohesive zone relationship of the multiple craze zone is found to be highly convex, which indicates effectiveness of rubber particletoughening. It is also observed that the effective peak traction, 7 MPa, in the crack-tipcohesive zone of HIPS (30% rubber content) is lower than the uniaxial yield stress of 9 MPa, presumably due to stress multi-axiality effects. In contrast, in PMMA, methanol localizes the crack-tip craze, weakening the craze traction for craze-void initiation toabout 9 MPa and the fibril pull-out stress to less than 6 MPa. This reduction in cohesive traction, coupled with a strongly concave traction-separation cohesive zone relationship,signifies environmental embrittlement of PMMA. These experimentally determined cohesive zone laws are compared with detailed numerical analyses of effective microscale-void growth ahead of a crack-tip in Part II.

Keywords: A. Fracture mechanisms, Crazing; B. Polymeric materials, Cohesive zone model; C. Optical interferometry, Inverse problem.

* Corresponding author. Tel.: +1-401-863-1456; Fax: +1-401-863-9009.

Email address: kim@engin.brown.edu (K.-S. Kim).

ManuscriptClick here to view linked References

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1 Introduction

Over many years cohesive zone models have been used to emulate material separation in fracture processes with a general view that cohesive strength and fracture energy are two important material parameters to describe macroscopic fracture behavior. However, it has been recognized recently that the functional shape of the cohesive zone traction-separation relation is sensitive to certain micromechanisms of fracture processes (see introduction of Hong and Kim, 2003). In addition, a critical step to measure the functional shape of the cohesive zone using a planar field projection method (P-FPM) has been introduced by Hong and Kim (2003), and Choi and Kim (2007). In this paper, we investigate two technically important mechanisms of polymer fracture – methanol embrittlement of polymethylmethacrylate (PMMA) and rubber particle dispersion toughening of high impact polystyrene (HIPS) – by measuring the cohesive zone traction-separation relation. The PMMA study is carried out to understand softening processes of craze void formation and fibril growth under a methanol environment, while the HIPS study is used to understand toughening processes caused by spatial dispersion of multiple inter-rubber-particle crazes. Crazing is widely regarded as the main precursor to fracture in glassy polymers due to its crack-like geometry and localized plastic deformation behavior. Experimental studies using electron microscopes as well as optical

interferometry (e.g. Weidmann and Döll, 1978; Kramer, 1983) have shown that a crazeconsists of a dense array of load-bearing fibrils separated by craze voids. Analytically, an individual craze has often been represented by the Dugdale model (Dugdale, 1960) which

assumes constant traction distributions within the craze zone during fibril pull-out (Döll,

1983; Williams, 1984). Our methodology, P-FPM, involves a combination of analytical, experimental and

numerical inverse approaches for measuring the cohesive zone laws uniquely from the elastic far-fields surrounding the crack-tip cohesive zone. Previous inverse-problem studies have derived the cohesive zone variables from either experimental measurements of the load - displacement or load - crack opening displacement (COD) curve data sets (Li et al., 1987; Bazant, 2002; Elices et al., 2002), or from whole-field deformation data obtained from optical measurements, such as moiré interferometry (Guo et al., 1999; Mohammed and Liechti, 2000; Tan et al., 2005). In both cases, a functional form of the cohesive zone law is assumed a priori, while the characteristic parameters of the functional form, such as the separation energy and maximum cohesive traction, are obtained by numerical fitting of the measured experimental data. As such, these inversion approaches are highly sensitive to the location of measurement and to measurement error(e.g. Burke, et al., 2007), and hence lack accuracy to assess the shape of the traction-separation relationship.

The planar field projection method uses a general form of cohesive crack-tip fields toobtain eigenfunction expansions of the plane elastostatic field in a complex variable representation. The general form is constructed by a shifted superposition of two non-singular linear elastic crack-tip fields. According to the general-form complex functions, the profiles of cohesive tractions and separation-gradients within a cohesive zone

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( )c x c , as depicted in Fig. 1, can be expressed in terms of two analytic complex functions ( )F x and ( )G x as follows;

2 1 22 12( ) ( ) ( ) ( ) ( )t x i t x x i x c x F x (1)

2 1 2 11

( ) ( ) ( ) ( ) ( )2

b x ib x u x i u x c x G xx

(2)

with indicating the difference between the values of the upper and the lower faces of the cohesive zone, where 3 4 for plane strain, (3 )/(1 ) for plane stress and and are shear modulus and Poisson’s ratio, respectively. Expressing

( )F x and ( )G x in terms of the Chebyshev polynomial of the second kind and applying analytic continuation, the cohesive tractions and the separation gradients can be represented with a set of cohesive crack-tip complex eigenfunctions which is complete and orthogonal in the sense of the interaction J-integrals at far field as well as at the cohesive zone faces. Then, the coefficients of the eigenfunctions in the J-orthogonal representation are extracted directly, using the interaction J-integrals at far field between the physical field and auxiliary probing fields. The separation profile within the cohesive zone is subsequently found by integrating,

2 1 2 1( ) ( ) ( ) ( )c

xx i x b ib d (3)

The parametric relationship between the tractions and the separations within the cohesive zone constitutes a cohesive zone law. The path-independence of the interaction J-integral enables us to identify the cohesive zone variables, i.e. cohesive tractions, separation-gradients and cohesive separations, as well as a cohesive zone law uniquely from the far-field elastic data.

In our study, we use the electronic speckle pattern interferometry (ESPI) to obtain the elastic displacement fields surrounding the crazing zone. A global numerical noise reduction algorithm is then applied to extract smooth equilibrium displacement fieldsfrom the experimentally measured displacement fields. The processed displacement fieldsare subsequently used in the analytical P-FPM to obtain the cohesive zone laws. The experimentally extracted cohesive zone laws will be compared with those from micromechanical analysis of effective microscale void growth ahead of a crack-tip in Part II.

2 Experimental measurement of crack-tip displacement fields

2.1 Material characteristics

Two different crazing processes are investigated in this paper: (i) crazing of steady-state methanol environmental crack growth in PMMA and (ii) multiple crazing in crack-growth toughening of HIPS. For PMMA subjected to mechanical loading in a methanol environment, an extended single craze zone develops ahead of a crack-tip. The methanol

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diffuses locally at this single craze zone causing plasticization of the fibrils, craze softening, and subsequent reduction of the material’s stress carrying capacity (Williams et al., 1974). In contrast, for rubber-modified HIPS, stress concentrations near cavitated rubber particles promote multiple crazing; the energy dissipation due to the generation and termination of these multiple crazes by rubber particles results in the high impact energy absorption of HIPS (Bucknall, 1977). Due to the formation of this multiple craze network in HIPS, the process zone of HIPS is broad and less well defined along the fracture plane. Hence the two cases studied here – single crazing zone in PMMA and multiple crazing in HIPS – represent two limits of the possible extent of crazing in polymers. For the purpose of discussion, the cavitated rubber particles in HIPS can be treated as voids, and are herein termed as micro-scale voids. To distinguish these micro-scale voids from voids that form within the craze zone (in both HIPS and PMMA), we term the latter as craze voids.

For both materials, single edge cracked bars are loaded in four point bending as shown in Figure 2(a). The PMMA bar specimen has dimensions of width

18.2 mmw , thickness 5.46 mmt and length 77.9 mmL . However, the specimen has end-milled cylindrical cut-outs of 8.97mm diameter centered at the crack-tip on both sides of the specimen, leaving the specimen thickness of only 1.14 mm around the crack-tip. This geometry is used to ensure near-plane-stress condition and absence of buckling for PMMA under environmental crack growth. For the PMMA specimen, we have used nominal values of Young’s modulus, 3.1 GPa and Poisson’s ratio, 0.35 to evaluate stresses from measured values of strains. The HIPS single edge cracked bar specimen has dimensions of width 10.0 mmw , thickness 5.10 mmt and length 60.0 mmL . HIPS is known to have volume fractions of rubber particles in polystyrene matrix in a range of 5 to 35 % (Katime et al.. 1995). The HIPS specimen in this experiment has rubber content of approximately 30% volume fraction and the morphology is shown in supplementary Figure S1. The particle size distribution is close to bimodal with primary and secondary diameters of 20 μm and 3 μm respectively. Wecarried out simple tension tests, as shown in supplementary Figure S2(a), for HIPS yielding at two different strain rates, and a simple relaxation test, to be compared with multiple crazing behavior in the crack-tip cohesive zone. The stress-strain curves at two strain rates, 0.18 and 0.89 micro strains per second, are shown in supplementary Figure S2(b). The curves exhibit upper yield stresses and lower yield (or flow) stresses. The lower yield stresses are approximately 1.5 MPa lower than corresponding upper yield stresses. Presumably due to the bimodal distribution of the rubber particle size, the upper yield stress, 13.2-14.5 MPa, of this HIPS specimen is substantially lower than the nominal value, 24 MPa, of typical HIPS reported in literature. The yielding behavior is close to that of HIPS 8350 reported in (Katime, et al.. 1995). The flow stress relaxes close to 8.4 MPa within two and a half hours in room temperature, as shown in supplementary Figure S2(c). The measured Young’s modulus of 23-25 GPa is at the higher end of the distribution 19 – 25 GPa for typical HIPS reported in MatWeb (http://www.matweb.com). Throughout this paper, we take Young’s modulus and Poisson’s ratio of the HIPS as 24 GPa and 0.35 respectively. In the following section, we describe how the displacement fields surrounding the crack-tip crazing zone are measured by the ESPI. The displacement fields are subsequently converted to strain fields and then to stress fields by multiplying the strain fields with the elastic stiffness of the medium.

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2.2 Optical measurement setup

Figure 2(b) shows a schematic of the four-beam phase-shifting ESPI used in this study (See also supplementary Figure S3(a) for our real physical setup). An expanded and collimated beam from a He-Ne laser ( 632.8 )nm is split into several diffracted beams by a transmitting diffraction grating G which has a square grid of 200pitches/mm. Among the diffracted beams, the first and zeroth order diffracted beams are collected by an achromatic doublet lens 1L , and focused onto an aperture plane A

located at the focal plane of the lens 1L . A special aperture denoted as A selects two first-order diffracted beams in the horizontal or vertical plane. By rotating the aperture A , two horizontal or two vertical illuminating beams can be chosen to measure displacement fields in both directions separately. Then, the selected beams are expanded and recollimated with an oblique illumination angle by another identical achromatic lens

2L . Scattered laser from a diffuse specimen surface is collected and imaged onto a CCD camera imaging system. Acquired video signals are sent to a frame grabber board of an IBM-PC where the phase calculations are performed. The CCD camera imaging system has 512x480 pixel resolution, corresponding to 15.4 m and 12.3 m in the horizontal and vertical directions, respectively. Aspect ratio of each pixel of the camera is 0.80 for the field of view of 7.9 mm x 6.9 mm.

The four-step phase-shifting method was employed by translating the diffraction grating G diagonally with a piezoelectric actuator. The piezoelectric actuator was controlled by a 12-bit D/A converter connected to the same computer. The piezoelectric actuator used in this study was calibrated by measuring shifts of moire interference fringes obtained by placing another diffraction grating identical to the first grating G at the location of specimen. Also, the displacement measurement sensitivity of the speckle interferometer was calibrated by measuring a known amount of rigid-body rotation of a diffuse sample. From this sensitivity calibration, a 2 phase fringe is found to represent displacement differences of 2.59 m in both horizontal and vertical directions. Although one fringe can be resolved into 255 grayscales, it is believed that the phase shifting speckle interferometer has a phase accuracy of about 2/100.

The ESPI was designed to measure displacement fields around a crack-tip in a four-point bending test specimen (Fig. 2(a)). The illuminating angle was reduced to about 7so that the interferometer has an increased measurement range within the field of view upto the unstable fracture of single-end-notched specimens. An incremental measurement scheme was used to trace the displacement fields of the surface points as load level increases by using a deformed configuration of current measurement step as a new reference configuration for the next measurement step. Rigid-body translations of the specimen during each loading step are compensated by using digital image correlation of modulation maps. Then, a direct superposition of incremental measurements was made to obtain cumulative total displacement fields.

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2.3 Experimental procedure and results

A four point bending fracture test specimen of 1.14 mm thick PMMA is kept in contact with an 80:20 mixture of methanol/water at room temperature during incremental loadings. The resulting displacement fields around an environmental craze are incrementally measured using the ESPI. In each loading increment, optical phase measurements are conducted after the relaxation of the environmental craze reaches a saturation point. The incremental phase change maps were added to construct cumulative wrapped displacement maps in each loading stage. The cumulative wrapped displacement fields in the horizontal and vertical directions at four different loading stages are shown in Fig. 3. Each 2 phase fringe represents the displacement difference of 2.59 m per our sensitivity calibrations. For purposes of comparison between a sharp crack-tip field and a cohesive crack-tip field, we show a typical unwrapped analytical KI displacement fieldwith contours of 2 phase in supplementary Figure S3(b). These KI field fringes are clearly observed in the ESPI phase images of a near crack-tip field shown in supplementary Figure S3(c). The phase image was taken with our ESPI from a four point bending experiment of a single edge cracked PMMA specimen without near-tip cut out to have 1.14 mm thickness; the specimen was loaded in an ambient atmosphere without methanol/water environment.

Crack-tip displacement fields in HIPS under mode I loading are also measured using the same test procedure as detailed above with 5.10 mm thick samples, but without subjecting HIPS to a methanol/water environment. The resulting cumulative wrapped displacement maps in HIPS at four incremental steps of loading are shown in Fig. 4. Observe that an elongated nonlinear deformation zone extends from the crack-tip, which is typical of localized cluster of multiple micro-crazing. Throughout this paper, we will call the localized cluster of multiple micro-crazes as “cohesive zone” of HIPS. Due to the large displacement gradient within this cohesive zone, the wrapped displacement maps in Fig. 4 show poor fringe visibility near the crack-tip. The 2 phase fringes show similar profiles to the analytical KI-displacement field until load step 4. Beyond this point, the wrapped displacement maps show large deviations from the analytical KI-displacement field. This can be attributed to the growth of the non-linear deformation zone of the craze(on the order of mm).

We use the cumulative displacement fields in the third step of both the PMMA andHIPS samples as input data for P-FPM. A global noise reduction algorithm is necessary to remove the remaining low-frequency noise in the experimental data prior to performing the inverse analysis. Formulations of the global numerical noise reductionalgorithms are discussed in Appendix. The equilibrium smoothing method was applied tothe measured experimental data set in order to extract the smooth equilibrium deformation fields. A linear elastic domain in the vicinity of but some distance away from a crack-tip is selected from the experimental field of view as depicted in the inset ofFig. 1. In this paper, we set a one-to-one correspondence between the experimental sampling points and the nodal points in the finite element mesh of the domain for equilibrium smoothing. An additional traction free condition was also imposed on the nodes along the crack faces.

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For the case of environmental crazing in PMMA, the wrapped displacement field shown in Fig. 3 is unwrapped by removing the 2 phase jumps. The unwrapped continuous displacement field around the environmental stress craze in PMMA containslow-frequency fluctuations due to experimental noise (Fig. 5). After applying the equilibrium smoothing method with boundary smoothing to the noisy field, a smooth equilibrium displacement field is obtained as shown in Fig. 6. Observe that a significant amount of the experimental noise is removed. Similarly, experimental noise in the measured displacement field in HIPS (Fig. 4) is also effectively removed by our equilibrium smoothing method. Compare the unwrapped displacement field around the non-linear fracture process zone in HIPS before and after equilibrium smoothing in Figs. 7 and 8 respectively.

Next, we evaluate the J-integral values (Rice, 1968) from the displacement fields of

both the PMMA and HIPS samples along several contours w (see inset in Fig. 1)

, 1 ,11

( )2

w

J w u n u n ds

(4)

where n is the outward normal to the contour w and , 1 or 2 . The J-integral values calculated from the unprocessed and processed experimental data are compared in Fig. 9. Observe that the J-integral values evaluated from the actual experimental data set along different contours exhibit large fluctuations (denoted by open squares). By contrast, J-integral values evaluated after equilibrium smoothing demonstrate their path-independence property. Comparison between the J-integral values evaluated using equilibrium smoothing with and without boundary smoothing (crosses and open circles respectively) show the former to have reduced errors near the domain’s boundary. The relative errors in tractions and separations can be seen as half the standard deviation of the J values relative to the path independent J value provided in Fig. 9; the relative errors in traction and separation are within ±5%, since the relative errors in J values are within ±10%. Further accuracy issues of P-FPM, such as the effects of the cohesive zone length,and the effects of measurement and numerical process errors, have been addressed in (Hong and Kim, 2003).

3 Cohesive zone laws for crazing

As aforementioned, our focus is to determine the cohesive zone laws for both single craze growth in PMMA in a methanol environment and multiple crazing in HIPS. These separate cases represent two limits of the possible crazing extent in polymers.

3.1 Crazing of steady-state environmental crack growth in PMMA

We first examine the cohesive zone law for methanol crazing in PMMA. The effective location of the crack-tip is estimated from the field of , using the 2-

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dimensional conservation M-integral (Knowles and Sternberg, 1972; Budiansky and

Rice, 1973)

, ,1

2M u n u n x ds

. (5)

Along with the J-integral, the equivalent position of a linear elastic crack-tip on a crack plane with respect to an arbitrary origin is given by

0

Mx

J (6)

The crack-tip position, together with the experimentally estimated cohesive zone size, and the regularized crack-tip displacement fields (Fig. 6) are used as input data for the inversion algorithm in Eqs. (1) – (3). We found that the extracted traction and separation distributions oscillate wildly as the extraction order of the Chebyshev polynomial N increases beyond N= 1 due to the ill-conditioning of the inverse problem with our pixel-matching grid-finite-element approximation for equilibrium soothing. As such, we limit our study to a two-term representation with N = 0 and 1. Thus, the extracted cohesive tractions and separations represent first-order approximations of the actual values.

Figures 10(a)-(c) show the profiles of cohesive zone variables, namely (a) cohesive traction, (b) separation-gradient and (c) cohesive separation, extracted from the smooth equilibrium field. The cohesive zone law, parametrically constructed from the traction and separation distributions within the cohesive zone, is displayed in Fig. 10(d). These data sets correspond to the displacement fields of the third step in Fig. 3, for which the crack has grown from the original length shown in the first step of Fig. 3. Observe that the extracted traction distribution undergoes rapid initial softening with negligible opening from 9 MPa to 6.5 MPa near the craze-tip. The peak stress 9 MPa is still small compared to the nominal mechanical tensile strength of 40-80 MPa, indicating that the critical traction of chemical environmental crazing at the craze-tip multiaxial stress state is substantially lower than the mechanical tensile strength. The resulting shape of the cohesive zone law for the methanol-attacked craze is highly concave, which is indicative of brittle-like failure behavior. We attribute this embrittlement to the methanolenvironment which decreases the local glass transition temperature of the craze in the process zone extending from the craze-tip. As a result, creep deformation of the fibrils is increased, and the fibril pull-out stress is reduced. Due to the limited diffusivity of the methanol in the PMMA, craze voiding or initiation of fibril pull-out to a large extent is confined to the craze-tip region. The accelerated near-tip craze voiding sustained by high concentration of methanol then leads to the rapid uninhibited drop in the cohesive traction.

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3.2 Multiple micro-crazing in crack-growth toughening of HIPS

The ductility of glassy polymers can be improved by rubber-toughening – the dispersed rubber particles (order of μm size) in the glassy polymer matrix act as multiple micro-craze initiation sites in the process zone ahead of the crack-tip. Kinloch and Younf (1983) showed that the fracture process zone in HIPS contained multiple parallel micro crazes between the rubber particles. Sauer and Chen (1983) further noted that multiple micro crazes in the rubber-toughened glassy polymer induce a fairly rough fracture surface with many flaky microstructures. Along this line, many experimental and numerical studies have also attempted to explain the rubber toughening mechanisms in

glassy polymers (e.g. Donald and Kramer, 1982; Argon and Cohen, 1990; Socrate et al.,2001).

Here, we examine the traction-separation distributions for multiple micro-crazing in the growth-toughening stage of rubber-toughened HIPS using the displacement fieldspresented in the third step in Fig. 4. The cohesive zone characteristics measured with the displacement fields represent those of under-developed cohesive zone in the middle of the toughening process. Similar to the earlier PMMA case, we adopt the two-termChebyshev polynomial representation of N = 0 and 1 in the inverse analysis. The extracted profiles of cohesive zone variables, namely (a) cohesive traction, (b) separation gradient and (c) cohesive separation, as well as (d) cohesive zone traction-separationrelation are shown in Fig. 11. These cohesive zone variables should be viewed as global effective quantities for multiple micro-crazing. For verification purposes, we note that the separation energy (area under the curve) in Fig. 11(d) is 96.7 J/m2; this value agrees well with the far-field J-integral value of 101.7 J/m2 in Fig. 9(b).

The cohesive zone law in Fig. 11(d) indicates that the maximum cohesive traction is about 7 MPa, which is lower than the measured “relaxed flow stress” of 8.4 MPa shown in supplementary Figure S2(c). This discrepancy can be attributed to high stress multi-axiality effects resulting from geometrical out-of-plane constraint of the HIPS specimen, and also viscoelastic effects on near-tip deformation which are neglected in our analysis. The damage in this under-developed cohesive zone consists of a mixture of multiplemicro-crazes and cavitated rubber particles. The corresponding shape of the traction-separation relation of the under-developed cohesive zone is distinctively convex. Notethat the shape of this cohesive zone law for multiple micro-crazing differs significantly from that of single methanol craze formation and growth in Fig. 10(d).

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4. Discussions

Cohesive zone constitutive laws of various functional forms – bilinear, linear-parabolic, exponential and trapezoidal – have been used to model the process zones of semi-brittle materials (e.g. Xia, et al., 2009). For the specific cases of polymer crazing we have studied, negligible contribution from the hardening portion of the cohesive zone law to the overall separation energy G was observed, as shown in Figs. 10(d) and 11(d). The softening regime of the cohesive zone law, however, can vary from concave to convex shape. As an indicator of the convexity of the traction-separation relationships, we define cohesive zone softening shape factor G/G0, where * *

0 / 2G is the linear softening

separation energy, and * and * the peak cohesive traction and total separation respectively. For example, the cohesive zone law for PMMA in methanol has G/G0 = 0.32, while HIPS has G/G0 = 1.36. Note that the cohesive zone law has a convex shape for G/G0 > 1 and a concave profile for G/G0 < 1. Our experiments indicates that the former is representative of multiple craze zone broadening, which enhances toughness within a given * , while the latter suggests embrittlement characterized by concentrated tractions at the craze-tip, trailed by rapid softening cohesive tractions behind the craze-tip.

The planar field projection method (Hong and Kim, 2003) used in this study is based on the 2-dimensional plane elasticity formalism. However the actual stress state in a specimen with a finite thickness is 3-dimensional. Hence, the proper interpretation of the extracted results from P-FPM requires some understanding of the nonlinear fracture processes under a 3-dimensional stress state. Nonlinear fracture processes in materials are governed by a competition among different localized deformation mechanisms, such as shear yielding, void growth, cleavage separation and chain scission. Depending on the atomic bonding and the intrinsic structures, the nonlinear fracture process is determined by the response of the material near a macroscopic crack-tip to different stress components, such as the Mises effective stress, the hydrostatic stress and the maximum principal stress.

The fracture processes in glassy polymers, which consist of frozen secondary bonds of non-crystalline covalently bonded long-chain molecules, are primarily governed by crazing and shear yielding. Two separate failure criteria are often used to explain the

nonlinear deformation mechanisms in the glassy polymers (Kinloch and Young, 1983): the shear yielding criterion which is governed by the effective shear stress component; and the craze initiation criterion which is governed by the hydrostatic stress component. Thus, the effects of 3-dimsionality on the nonlinear fracture process zone in a finite thickness specimen can be explained by comparing the effective shear stress and the hydrostatic stress in the plane-strain or the plane-stress crack-tip fields. Shear yielding is promoted near the free surfaces in ductile materials. On the other hand, the crazing initiated by the hydrostatic stress state is more likely to grow inside of the specimen. Thus, the craze zone formed in the finite thickness specimen usually has a curved craze front on the crack plane. Therefore, the cohesive zone law extracted from the surface measurement data must be interpreted as an average cohesive zone law in the thickness direction or the plane-stress cohesive zone law, depending on the size of the cohesive zone and the field of view with respect to the thickness of the specimen. In turn, we

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should consider the effects of the 3-dimensionality for specimen design in experimental measurements in order to provide the correct interpretation of the P-FPM results.

The accuracy of the extracted cohesive zone variables is limited by several factors. The first is the inversion distance, which can be normalized by the size of the cohesive zone. Other important parameters include the error and spatial resolution of the experimental data, the numerical error retained in the equilibrium smoothing with finite element approximation, and the cut-off polynomial order N of the cohesive zone variables. Graded meshing instead of the uniform grid meshing of the finite element equilibrium smoothing is believed to increase the cut-off polynomial order. For further studies to improve the accuracy, optimal experimental and numerical parameters need to be obtained from the error analysis on the experimental/numerical inversion processes.

5. Concluding remarks

In this paper, a hybrid framework for an inverse analysis of the crack-tip cohesive zone model was developed and applied to measure the cohesive zone laws of two nonlinear fracture processes in glassy polymers, namely single craze zone formation during steady-state crack growth in PMMA in methanol environment and multiple craze zone broadening in crack growth toughening of HIPS. An analytical solution method of the inverse problem was employed to extract cohesive zone laws from elastic far-fields surrounding a crack-tip cohesive zone. The electronic speckle pattern interferometry (ESPI) with phase-shifting technique was used to measure the full-field crack-tip displacement near the fracture process zones in PMMA and HIPS. This measured displacement field was then processed by a numerical noise reduction algorithm to remove the experimental noises. The obtained smooth equilibrium field was subsequently used as input data in the planar field projection method to extract the cohesive zone laws.

With the above framework, results show that in PMMA, methanol localizes the crack-tip craze, weakening the craze traction for craze void initiation to about 9 MPa and the fibril pull out stress to less than 6 MPa. This reduction in cohesive traction, coupled with a strongly concave traction-separation cohesive zone relationship, suggests environmental embrittlement of PMMA. By contrast in HIPS, multiple craze formations initiating from and terminating at cavitated rubber particles result in a broadened craze zone ahead of a crack-tip under mechanical loading. The corresponding cohesive zone relationship of the multiple craze zone is found to be highly convex, which indicates effectiveness of rubber particle toughening. It is also observed that the effective peak traction, 7 MPa, of the crack-tip cohesive zone in HIPS (30% rubber content) is lower than the uniaxial yield stress of 9 MPa, presumably due to effects of near crack-tip stress multi-axiality and nonuniform rate of viscoelastic relaxation in the near-tip deformation field.

As the planar field projection method can measure the functional shapes of cohesive zone traction-separation relations, we have defined the cohesive zone softening shape factor G/G0 as an indicator of fracture process patterns in the cohesive zone. Here, G is the overall separation energy, * *

0 / 2G the linear softening separation energy, and * and * the peak cohesive traction and total separation respectively. A convex

functional shape of the cohesive zone traction-separation relation (G/G0 > 1) indicates

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toughness enhancement for a given total separation, * , while a concave profile (G/G0 < 1) suggests embrittlement characterized by traction concentration at the craze-tip, trailed by rapidly decaying cohesive tractions behind the craze-tip. In our experiments, it is found that the cohesive zone law for PMMA in methanol has G/G0 = 0.32, while HIPShas G/G0 = 1.36.

These experimental observations are compared, in Part II, with detailed numerical simulations in which the volume expansion caused by multiple crazes is partitioned and regarded as part of the volume expansion in an effective micro-scale void growth. In addition, the embrittlement mechanism of methanol craze in PMMA in which the softening agent diffuses from the crack surface is also compared in Part II with the embrittlement mechanism caused by void pressure in which the pressure inducing vapor diffuses mostly in the bulk. In both cases, the softening shape factor G/G0 is found to be a useful parameter for quantitative comparisons. In turn, the planar field projection method is clearly verified to be a robust tool to bridge micro-mechanisms of fracture processes and macroscopic fracture characteristics in both experimental and modeling studies.

Acknowledgements

The support of this work by the MRSEC Program of the National Science Foundation

under Award Number DMR-0079964 and DMR-0520651, and by the National

University of Singapore under the Research Grant NUS-6-32254 is gratefully acknowledged. The authors also thank Dr. Shuman Xia for helping HBC to carry out tension tests and characterization of HIPS specimens.

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15

Appendix A: Numerical noise reduction in full-field experimental data

One defining characteristics of inverse problems, such as P-FPM, is ill-conditioning, which makes the inverse solution highly sensitive to the input data. Therefore, an efficient noise reduction algorithm detailed below is required to remove the inevitable experimental noise prior to performing the inverse analysis.

A.1 Equilibrium smoothing method

The global numerical noise reduction in experimentally measured displacement fields, called equilibrium smoothing, is a problem of finding a displacement field that is closest to the measured field among possible equilibrium fields satisfying the governing continuum field equations. Consider an elastostatic problem on a domain , in which displacement fields are denoted by ( )iu x at x . The corresponding strain field ij

and stress field ij are defined by the following equations using stiffness tensor ijklC

, ,( )/2ij i j j iu u (A.1)

ij ijkl klC (A.2)

where ,i ju denotes the partial derivatives of iu in jx direction and the summation

convention is assumed for the repeated indices. The governing field equation for elastostatics is the equilibrium equation,

, 0ij j ib (A.3)

where ib denotes body force acting on .Let ( )iu x be an experimentally measured displacement field of the elastostatic

problem in the domain . The measured displacement field ( )iu x includes an unknown amount of experimental error ( )i x and does not necessarily satisfy the governing field equations

( ) ( ) ( )i i iu u x x x (A.4)

Error in the experimental data can be quantitatively characterized by using Euclidian norm of error.

The global numerical noise reduction problem can be considered as a PDE-constrained optimization problem which attempts to minimize the Euclidian error norm in the least-square sense while enforcing the equilibrium conditions, i.e.

21minimize ( )

2i i iE u u u d

x x (A.5)

,subject to 0 in ij j ib (A.6)

16

Implementing (A.5) and (A.6) for numerical analysis, the displacement fields ( )iu x

and ( )iu x can be approximated by the finite element interpolation functions ( )( )kM x and ( )( )kN x in the finite element mesh and the data sampling mesh as follows

( )( )

1

( ) ( )m

kki

k

u M u

x x (A.7)

( )( )

1

( ) ( )n

kki

k

u N u

x x (A.8)

where ( )ku and ( )ku denote arrays of nodal displacement values in the finite element mesh and the data sampling mesh, and m and n are the total degrees of freedom in the respective meshes. By inserting (A.7) and (A.8) into (A.5), a symbolic representation of the discrete error minimization problem can be formed as,

1 1minimize [ ]

2 2E u u Pu u Qu u R u

(A.9)

where ( ) ( )( ) ( )i jij m m

P M M d

P x x

( ) ( )( ) ( )i jij m n

Q M N d

Q x x

( ) ( )( ) ( )i jij n n

R N N d

R x x

and u and u are the arrays of nodal displacement values in the equilibrium field and the measured data. Then, the equilibrium constraints in linear elasticity can be easily implemented into a linear algebraic form using standard finite element formulations. Neglecting body forces, a linear algebraic form of the equilibrium equations can be written as,

b b11 12

i12 22

u pK K

u 0K K(A.10)

where u and p represent arrays of nodal displacements and nodal forces, and subscripts b and i indicate boundary and internal nodes, respectively. The nodal forces on the internal nodes vanish from equilibrium conditions. Because the nodal forces on the boundary are unknown in the equilibrium smoothing, the equilibrium constraints can only be applied to the interior nodes. A linear algebraic form of the equilibrium constraints in the internal domain can be written as,

Ku 0 (A.11)

where K is defined by

12 22 K K K (A.12)

Note that the stiffness matrix K in (A.12) is not a square matrix.

17

Finally, the constrained minimization problem in (A.5) and (A.6) is converted into a discretized form as follows

1 1minimize [ ]

2 2E u u Pu u Qu u R u

(A.13)

subject to Ku 0 (A.14)

The equilibrium constraints in (A.14) are then incorporated into a modified error

functional by using the Lagrange multiplier method (Bertsekas, 1982) as1 1

minimize [ , ]2 2

E u u Pu u Qu u R u Ku (A.15)

where is an array of Lagrange multipliers. Therefore, the constrained minimization problem becomes a simple minimization problem with increased degree of freedom shown as

TE

E

Pu Qu K 0u

Ku 0

(A.16)

Equation (A.16) can be represented with a partitioned matrix and vectors as follows,

T

QuP K u

K 0 0

(A.17)

Using the inversion formula of partitioned matrix (Horn and Johnson, 1985), Eq. (A.17)can be solved as follows,

1 1 1 1 1 1

1 1 1

P P K KP P K Quu

KP 0

(A.18)

where 1 KP K .Finally, we obtain the optimized displacement field u without any subjective choice

of smoothing parameters,

1 1 1 1 u P P K KP Qu (A.19)

The general equilibrium smoothing formula in (A.19) can be used to find a smooth equilibrium field on an arbitrary finite element mesh regardless of geometries and sampling points in the experimental data.

Many whole-field optical measurement techniques, such as ESPI, moire interferometry and digital image correlation technique, provide measured experimental data in uniform square or rectangular grids owing to electronic image acquisitions and digital data processing systems. Consider the discrete sampling points in the uniform square grids as nodal points in a 2D finite element mesh and using an alternative error functional,

18

21[ , ]

2E u u u Ku (A.20)

a simplified form of the equilibrium smoothing formula in (A.19) can be found as

1( ) u I K KK K u (A.21)

Equation (A.21) can be solved numerically using the conjugate gradient method for large systems of linear algebraic equations.

A.2 Equilibrium smoothing method with boundary smoothing

Although significant amount of experimental noise can be removed in the interior of the domain using the interior equilibrium smoothing algorithm described above, some errors still reside on the boundary of the domain. A more refined error reduction scheme is required to remove residual noises on the boundary. A modified constrained optimization problem which includes a penalty function term of cubic-spline smoothing along the boundary is given by

22

22

1 1minimize ( ), ,

2 2i

i i iu

E u u u d d ss

x (A.22)

,subject to 0 in ij j ib (A.23)

where is a boundary smoothing coefficient. Using a finite difference formulation of the cubic-spline penalty function on the boundary and the finite element formulation of equilibrium constraint in the interior domain, a modified constrained optimization problem is represented in a discretized form,

22b

1minimize [ , ; ]

2 2E

u u u Cu Ku (A.24)

where a matrix C denotes a discrete approximation of the second derivative operator along the boundary. By minimizing the error functional in (A.24), we obtain

12 bb

22 i i

12 22

I C C 0 K uu

0 I K u u

0K K 0

(A.25)

The system of linear equations in (A.25) is represented with a partitioned matrix and vectors as

19

T

u uA K

0K 0

(A.26)

where

I C C 0A

0 I(A.27)

Finally, the optimized displacement field can be found as,

1 1 1 1 u A A K KA u (A.28)

where Δ -1 KA K (A.29)

The inverse of the partitioned matrix A can be obtained by inverting only the small matrix ( ) I C C . Assuming that the experimental noise level is approximately uniform throughout the entire domain, an optimal boundary smoothing parameter can be objectively found by solving (A.28) iteratively while changing until the Euclidian norm of error reduction on the boundary reaches the same value as the interior of the domain.

Figure Captions

Figure 1. Mathematical model of a crack with a cohesive zone of size 2c, with traction t(x) and separation δ(x) in the cohesive zone in an elastic body of shear modulus μ and Poisson’s ratio ν. Inset shows a domain near the crack-tip, with a contour integral path Γw surrounding the crack-tip.

Figure 2. (a) Schematic of the four-point bending test. (b) Four-beam phase-shifting speckle interferometer setup to measure two in-plane displacement components with a cross diffraction grating G, two lenses L1, L2 and a mirror on the aperture plane A.

Figure 3. Cumulative wrapped displacement maps of horizontal and vertical displacement components near a crack-tip in PMMA under four incremental loadings.

Figure 4. Cumulative wrapped displacement maps of horizontal and vertical displacement components near a crack-tip in HIPS under four incremental loadings.

Figure 5. Noisy unwrapped displacement fields around a crack-tip in PMMA.

Figure 6. Smooth equilibrium displacement fields with boundary smoothing around a crack-tip in PMMA.

Figure 7. Noisy unwrapped displacement fields around a crack-tip in HIPS.

Figure 8. Smooth equilibrium displacement fields with boundary smoothing around a crack-tip in HIPS.

Figure 9. J-integral values calculated along various contours in (a) PMMA and (b) HIPS. Open squares for J-integral values before equilibrium smoothing; open circles for J-integral values after equilibrium smoothing without boundary smoothing; crosses for J-integral values after equilibrium smoothing with boundary smoothing.

Figure 10. Experimentally extracted cohesive zone variables and a cohesive zone law for crazing of steady-state environmental crack growth in PMMA.

Figure 11. Experimentally extracted cohesive zone variables and a cohesive zone law for multiple micro-crazing in crack-growth toughening of HIPS.

Figure 1 to 8 with figure captions

The cat on the moon

Figure 1

(a)

(b)

Figure 2

P/2 P/2

P/2P/2

Lw

Horizontal Vertical

Figure 3

1

2

3

4

Horizontal Vertical

Figure 4

1

2

3

4

(a) 1( , )u x y (b) 2( , )u x y

(c) 1( , )u x y (d) 2( , )u x y

Figure 5

(a) 1( , )u x y (b) 2( , )u x y

(c) 1( , )u x y (d) 2( , )u x y

Figure 6

(a) (b)

(c) (d)

Figure 7

(a) (b)

(c) (d)

Figure 8

Fig

ure

9

Fig

ure

10a

b

Fig

ure

10c

d

Fig

ure

11a

b

Fig

ure

11c

d

Supplementary Figures

Figure S1. Morphology of rubber particles in HIPS.

Figure S2. (a) Uniaxial stress-strain tensile testing of HIPS specimen. (b) Uniaxial stress-strain curves for HIPS samples loaded at unaixial strain rates of 5 18.9 10 s and

5 11.8 10 s respectively. (c) Stress relaxation profile for HIPS sample loaded previously at 5 11.8 10 s at fixed strain of ε = 0.04.

Figure S3. (a) Optical setup of four-beam phase-shifting speckle interferometer: symbols G, A, L1 and L2 represent the cross diffraction grating, mirror on the aperture plane, and two lenses. (b) Analytical K-field induced horizontal and vertical displacement contoursnear a crack-tip. (c) Experimental cumulative wrapped displacement maps of horizontal and vertical displacement components near a crack-tip in PMMA (absence of methanol environment); the enclosed dotted region in (c) denotes K-field dominant zone as shown by similarity in contour profiles with analytical solution in (b).

Supplementary Material Fig s1 and Fig s2a

Figure S1

100 μm

Figure S2a

Su

pp

lem

enta

ry M

ater

ial F

ig s

2bc

Figure S3a

Figure S3b

G

A

L1

L2

CCDspecimen

Horizontal Vertical

Supplementary Material Fig s3

Figure S3c

Horizontal Vertical