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Measurement of the J integral in polycarbonate using the method of caustics.


Some noncrosslinked polymers are capable of sustaining very large inelastic deformations before eventual failure. The micromechanism responsible for such behavior can be described in terms of the generation of molecular orientation in the initially randomly oriented polymer. These mechanisms are rate dependent in nature; typically, at high rates, a brittle elastic behavior is observed, while at lower rates, viscoplastic behavior can be observed. The constitutive behavior of these polymers are characterized using traditional elastic, viscoelastic or viscoplastic theories to varying degrees of success. The yielding and deformation localization behavior in polycarbonate has been examined by several investigators using different models. Caddell et al. (1) and Carpellucci and Yee (2) examined the pressure dependence of the yield behavior and postulated a modified Von Mises yield criterion for polycarbonate. The deformation localization behavior under homogeneous fields has been examined by Wu and Turner (3) and the necking and neck propagation has been studied extensively. Buisson and Ravi-Chandar (4) investigated the true stress-true strain behavior through combined photoelasticity for stress measurement and the grid technique for strain measurement: their results indicated that true material softening does not occur. The deformation localization behavior in the biaxial field near a crack tip was investigated by Donald and Kramer (5). Brinson (6) examined the ductile fracture behavior of polycarbonate using photoelasticity; detailed measurements of the localized deformation zone near the crack tip were also made and interpreted in terms of Dugdale model. Parvin and Williams (7) explored the ductile and brittle transitions as a function of the rate of loading and the thickness of the specimen and interpreted the transitions as a plane stress to plane strain transition. Sehanobish et al. (8) also examined the ductile to brittle transition and used a statistical approach relating the probability of ductile or brittle failure with the distribution of defect size. Wert et al. (9) examined the applicability of a modified J parameter as a fracture characterizing parameter and obtained J resistance curves which indicated significant increase in the J integral for small amounts of crack extension. In the present paper, we examine the nonlinear fracture behavior of polycarbonate and in particular address the feasibility of characterizing the fracture through the J integral as well as its measurement using the optical method of reflected caustics.


Polycarbonate exhibits complex nonlinear material behavior; in a simple uniaxial test, depending on the loading rate, brittle elastic behavior or ductile elastic-viscoplastic behavior can be exhibited. Also, deformation localization occurs leading to the formation of Luder's bands followed by stable necking and neck propagation. A number of investigations have been performed to examine these aspects and it is not our intent to review them here. We only describe the material properties that are relevant to the determination of the J integral through caustics. The load-displacement variation from a uniaxial tensile specimen is shown in Fig. 1, for three different loading rates. From these it can be seen that the material behaves linearly up to a load level of about 4 kN (corresponding to a stress of about 40 MPa), becomes nonlinear and at some point has a horizontal slope indicating an instability. At this load shear bands form which eventually grow into a stable neck profile which then grows along the entire length of the specimen at nearly constant load (called cold drawing). Significant influence of the rate of loading is observed only on the peak load at onset of instability and the subsequent load drop, but not in the onset of nonlinearity or in the steady state neck propagation load. In the biaxial stress field near a crack tip the geometric constraint does not permit large scale drawing and thus very little deformation localization occurs in the early stages of crack growth (as discussed in the next section). The region of the stress-strain curve that is of interest is the segment before the peak load; this can be described by a power law model. For the low load rates, describing the constitutive behavior by

[Epsilon]/[[Epsilon].sub.0] = [Sigma]/[[Sigma].sub.0] for [Sigma] [less than] [[Sigma].sub.0]

= [([Sigma]/[[Sigma].sub.0]).sup.n] for [Sigma] [greater than] [[Sigma].sub.0] (1)

and fitting to the experiments, we found n = 2.69, [[Epsilon].sub.0] = 0.0172, [[Sigma].sub.0] = 41.38 NPa and E = 2.4 GPa. While it is the usual practice to consider the peak in the nominal stress strain curve as the yield stress, we have used the 0.2% proof stress as the definition for the yield stress [[Sigma].sub.0]. The crack tip stress and deformation fields will thus be interpreted in terms of an HRR field.


The rate dependent material behavior of polycarbonate becomes apparent in its fracture behavior. By varying the loading rate over a small range, the fracture behavior can be changed from a ductile fracture mechanism to a brittle fracture mechanism. To examine the rate dependence, fracture experiments were performed on standard compact tension (CT) specimens complying with the ASTM E-813-81 standard. A sharp crack was made by fatigue cycling until a 4-mm crack was extended. The load to load-point-displacement relations are shown in Figs. 2 and 3 for fast and slow loading rates respectively. In the case of fast loading, a brittle (unstable) fracture was observed with an abrupt drop in the load. Crack propagation was fast; from other similar experiments on this material (10), it is known that the crack propagates at speeds of about 500 m/s. As the loading rate is lowered, an abrupt transition occurs in the failure mechanism. At slow loading rates, slow stable crack growth governed by ductile fracture mechanism is initiated. In the early stages of crack growth, small localized deformation zones (LDZs) occur; these appear to be similar to the Luder's bands observed in the simple tension tests. As the loading progresses further, these deformation zones grow further; in our implementation of the method of caustics to the evaluation of the J integral, measurements were performed only when the LDZs were small compared to the extent of the fatigue precrack and furthermore, the measurements were performed well outside the LDZs. Given that the material is assumed to be governed by a power law relationship, it is possible to interpret the load to load-point-displacements in terms of the J integral; however, in the present investigation, we perform a direct measurement of the local deformation field to obtain the J integral.


The method of caustics or shadow spots has found widespread use in experimental fracture mechanics, notably in dynamic situations, perhaps because of the ease with which data may be gathered, interpreted and analyzed. Originally interpreted for Mode-I elastostatic problems (11, 12), the method has been extended to cases of dynamic crack propagation (13, 14), mixed-mode loading (15), and elastic-plastic problems (16, 17). The main idea in the interpretation of the method of caustics is that a particular asymptotic stress and deformation field that is characterized by a typical intensity parameter is assumed to exist near the crack tip. For pure Mode I or Mode II loading, this intensity parameter fully characterizes the field, but in the case of combined or mixed-mode loading, an additional parameter describing the mode mix needs to be determined. In the case of purely elastic deformations, the two characterizing parameters are [K.sub.I] and [K.sub.II]. In elastic-plastic problems, the characterizing parameters would be the J integral and the mode mix parameter [M.sup.p]. In the present paper, we describe the application of the method of caustics to the determination of the J integral under Mode I conditions in polycarbonate. The analytical tools necessary for determining J and [M.sup.p] under combined mode loading are in place (17), and the corresponding experiments are currently in progress.

Principle of Formation of Caustics

Consider a light ray traveling in the [-x.sub.3] direction, as illustrated in Fig. 4. The cracked plate is placed in the path of this light beam. The light ray strikes the specularly reflecting front surface of the specimen and reflects from it. The angle of reflection is determined by the local gradient of the surface, which in turn is governed by the deformation of the plate. If the surface deformation is known, the reflection process can be completely characterized. On the other hand, the inverse problem consists of making some observations on the reflected light field and then attempting to reconstruct the nature of the deformed surface. If the deformed surface is given by [x.sub.3] = -f(r, [Theta]), then for small surface deformations, the mapping of the light rays from the specimen plane ([x.sub.1], [x.sub.2]) to the virtual screen plane ([X.sub.1], [X.sub.2]) can be written as

[X.sub.[Alpha]] = [x.sub.[Alpha]] - 2 [z.sub.0] [Delta]f/[Delta][x.sub.[Alpha]] (2)

where [z.sub.0] is the distance between the specimen front surface and the virtual screen; [Alpha] has the range 1, 2. If the deformed surface has a steep gradient, the virtual extensions of all the light rays might form an envelope in space where no reflected light ray is observed. The bounding surface separating the regions where the light rays are observed and not observed is called the caustic surface. If a virtual screen is placed at a distance [z.sub.0] from the specimen front surface, a dark region called the shadow-spot, surrounded by a bright curve called the caustic curve is observed. The mathematical condition for the caustic surface to exist is that the Jacobian determinant of the mapping equation be zero. Thus,

[Delta]([X.sub.1], [X.sub.2])/[Delta](r, [Theta]) = 0 (3)

This equation, when evaluated for a given surface gives the locus of all points on the specimen surface where the Jacobian vanishes; this curve is called the initial curve. By an examination of the mapping, it can be shown that light rays that reflect from the specimen both inside as well as outside the initial curve are mapped outside the caustic curve on the screen plane and that the rays reflecting from the initial curve are mapped onto the caustic curve. Thus, the equation for the caustic curve is obtained by substituting Eq 3 into Eq 2.

Elastic Deformations

The deformed surface near the crack tip can be modeled using the standard asymptotic fields for either a linear elastic or a nonlinear power law material. We shall provide here only the brief results; the details can be found in the references listed above. For a linear elastic material, the displacement field near the crack tip is given by

[u.sub.3](r, [Theta]) = - v d[K.sub.I]/E[square root of 2[Pi]r] cos [Theta]/2 + v d[K.sub.II]/E[square root of [Pi]r] sin [Theta]/2 (4)

where v is the Poisson's ratio, E is the modulus of elasticity, d is the plate thickness and (r, [Theta]) are the polar coordinates centered at the crack tip. Letting f([x.sub.1], [x.sub.2]) = [u.sub.3](r, [Theta]) in Eq 2 and Eq 3, the caustic curve can be obtained. For Mode-I problems, the maximum dimension of the caustic transverse to the crack (called the transverse diameter) D can be related to the stress intensity factor as

[K.sub.I] = 2[square root of 2[Pi]]E/3v dz [(D/3.17).sup.2.5] (5)

Thus, by measuring the caustic transverse diameter, the stress intensity factor can be determined. The procedure for mixed mode problems is similar as described by Theocaris and Gdoutos (15).

Inelastic Deformations

Considering the material to be described by a Ramberg-Osgood model, the stress, strain and displacement fields near the crack tip for plane stress or plane strain can then be obtained (18-20). In particular, for plane stress, the [u.sub.3](r, [Theta]) displacement, which governs the shape of the reflecting surface, under combined mode loading, takes the form:

[u.sub.3](r, [Theta]) = [

.sup.p] r).sup.n/(n + 1)] [Psi]([Theta]) (6)

where [Mathematical Expression Omitted], n is the hardening exponent, [Alpha] is the Ramberg-Osgood parameter, and [[Epsilon].sub.0], [[Sigma].sub.0] are the yield strain and yield stress respectively. J is the path independent contour integral (21), [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are the angular variations of the strain components and [I.sub.n]([M.sup.p]) is dependent on the hardening exponent n and [M.sup.p], the latter being the plastic mixity factor defined (20) by

[Mathematical Expression Omitted]

For the case of pure Mode I, [M.sup.p] = 1 and for pure Mode II, [M.sup.p] = 0. Combined modes have values in between this range. Thus the out-of-plane displacement field of the surface as well as the stress and strain fields of the plate are completely characterized in terms of the two parameters J and [M.sup.p]. It is assumed that the elastic strains in the neighborhood of the crack tip are negligible. For the mapping in Eq 2, we now use f([x.sub.1], [x.sub.2]) = [u.sub.3](r, [Theta]) given by Eq 6 to determine the shape of the caustic curve; the shape and size of the caustic curve can then be used to determine J and [M.sup.p]. This amounts to assuming that the asymptotic field given in Eq 6 has a range of dominance in the region over which the initial curve is expected to lie.

Using the above crack tip stress field for a mixed-mode crack, the caustic curves and the initial curves can be evaluated. In order to determine d and [M.sup.p]. Zhang and Ravi-Chandar (17) proposed using two geometric measures; the maximum diameter of the caustic [D.sub.m], and its angular orientation [[Theta].sub.m], can be used to determine J and [M.sup.p]. From the geometry of the caustic, the following relationship between [D.sub.m] and the J integral can be obtained:

J = [S.sub.n]([M.sup.p])[Alpha][[Epsilon].sub.0][[Sigma].sub.0] [(1/[Alpha][[Epsilon].sub.0][z.sub.0]d).sup.(n + 1)/n] [([D.sub.m].sup.(3n + 2)/n] (8)

where [S.sub.n]([M.sup.p]) depends only on n and [M.sup.p]; Zhang and Ravi-Chandar (17) provide a plot of this dependence. [[Theta].sub.m] depends only on [M.sup.p]. Thus by measuring [[Theta].sub.m], [M.sup.p] is obtained first and then by measuring the maximum diameter [D.sub.m] of the caustic, J can be obtained by using Eq 8. Note that while the above analysis assumes that the out-of-plane displacement near crack tip is given by Eq 6, the proportionality of J to [([D.sub.m]).sup.(3n + 2)/n] can be obtained directly from the path independence of the J integral and the assumed constitutive model in Eq 1; the details of the deformation field affect only the magnitude of [S.sub.n]([M.sup.p]). In this paper, we examine the application of the above technique to nonlinear fracture characterization of polycarbonate.


We now turn to the experimental determination of the J integral using the method of caustics. With a view toward addressing combined mode problems, the compact tension-shear specimen introduced by Buchholz et al. (22) was selected for these investigations. A sketch of this specimen is given in Fig. 6. A slit was machined to nearly half the width of the specimen and then a fatigue crack was grown for about the last 4 mm. The stress intensity factors [K.sub.I] and [K.sub.II] for this geometry are given as (22):

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where [Alpha] is the angle between the normal to the crack and the load line. Under conditions of small scale yielding the J integral can be obtained as

[Mathematical Expression Omitted]

assuming plane stress conditions to be valid for the 5.6-mm-thick plate specimen. Thus by monitoring the applied load F, the theoretical estimates for [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [J.sup.Calculated] can be determined.

The specimen and loading attachments were placed on a table model Instron test frame; for the implementation of the method of reflected caustics, the test frame was placed next to an optical table. The optical arrangements for the monitoring of the reflected caustic patterns is shown in Fig. 5. The light beam from the laser is expanded and collimated to a parallel beam of light with 50-mm field of illumination. This beam falls on a cube beam splitter and one half of the initial intensity continues on to the specimen surface. Typically, reflections from the front and back surfaces are observed, but the back surface was painted flat black thus eliminating the back surface reflection. No special surface preparation was needed for the front surface; the commercially obtained GE Lexan polycarbonate had sufficiently smooth surface finish to be used without even a reflecting coating. The reflected light falls on the cube beam splitter and one half of the reflected light intensity is collected by a lens that projects the virtual image at the distance [z.sub.0] behind the specimen on to a real screen from which measurements are made; in practice, the image on this screen is monitored using a video camera and measurements are made at larger magnification to reduce the error in length measurements. The collecting optics were arranged such that the reference distance [z.sub.0] was 1 m.

The loading grips were positioned to apply a pure Mode-I load on the specimen and the specimen was loaded at a very slow crosshead rate (0.25 mm/min); the loading was interrupted at increments of about the loading was interrupted at increments of about 220 N and the resulting caustic pattern was captured on videotape for analysis of the J integral. The loading was continued until the localized deformation zone near the crack tip was about 4 mm long; at this point this zone is of the same size as the position of the initial curve and interpretation of the caustic data becomes difficult. The measured caustics can be interpreted either in terms of the linear analysis (using Eq 5) indicating the imposition of a K-dominant field near the crack tip or in terms of a nonlinear analysis (using Eq 8) indicating the assumptions of a J-dominant (HRR) field near the crack tip. The results of both interpretations are shown in Figs. 7 and 8. At low load levels, below about 1 kN, the interpretation of the caustics in terms of a K-dominant analysis provides a correct estimate of the stress intensity factor. Thus, the stress and deformation fields near the crack tip appear to be governed by the inverse square root singularity. On the other hand, for loads larger than 1 kN, the estimate of the stress intensity factor obtained from the linear elastic interpretation of caustics does not follow the theoretically calculated values of the stress intensity factor indicating that the near tip stress and deformation fields are not governed by the inverse square root singular field. The initial curve corresponding to these measurements lies inside the plastic zone and therefore the interpretation has to be based on the elastic-plastic deformation fields. It is possible, by varying [z.sub.0], to make the radius of the initial curve larger and thus interpret the caustics using the K-field analysis; however, since our interest here is to demonstrate that the J integral can be measured directly, this was not attempted.

From the measured caustic diameters, the J integral was computed using Eq 8. The variation of the measured value of J is compared with the value of J calculated using Eq 10 as a function of the applied load. The maximum deviation between the measured and calculated values of J is about 10%; clearly the agreement is better than that observed with the linear analysis. This indicates that the crack tip stress and deformation fields could be interpreted in terms of the HRR field. Note that as the loading progressed, the crack experienced stable crack growth over a length of about 4 mm, with the initiation occurring at [Mathematical Expression Omitted]. This corresponds to the [K.sub.IC] value obtained with the high rate tests on the CT specimens. The measurement of the position of the caustic should provide a measure of crack extension as well, but this was not very precise; a direct measurement of the crack position (with a microscope for a magnified view of the crack tip) is desirable and this work is in progress. However, even without the precise crack positions known, it is possible to get some idea about the increase in the crack growth resistance of polycarbonate. For a very small crack extension, the J integral has increased by almost an order of magnitude indicating the significant influence of the localized deformation zone on the resistance to crack growth.


The fracture behavior of polycarbonate was examined. As is well known, polycarbonate undergoes a transition from ductile to brittle fracture at a certain loading rate. The value of [K.sub.IC] was measured from the brittle fracture experiments. For the case of ductile crack growth, the feasibility of measuring the d integral using the optical method of reflected caustics was examined. It was found that the method of caustics was indeed capable of reliable determination of the J integral; the value of J at instantiation correlates well with the value of [K.sub.IC]. Furthermore, during slow stable crack extension, the resistance of polycarbonate to crack growth increases almost by an order of magnitude for a crack extension of about 4 mm. Further work on the mixed mode loading effects is in progress.


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A. DHUMNE and K. RAVI-CHANDAR, Department of Mechanical Engineering, University of Houston, Houston, Texas 77204-4792
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Author:Dhumne, A.; Ravi-Chandar, K.
Publication:Polymer Engineering and Science
Date:Mar 15, 1995
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