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Polymer Journal, Vol. 8, No. 2, pp 181-189 (1976) Fracture Processes in Ductile Polymers. I. Localized Plastic Deformations of Polycarbonate and Nylon Films lkuo NARISAWA, Masaru ISHIKAWA, and Hiroyuki OGAWA Faculty of Engineering, Yamagata University, Jonan 4-3-16, Yonezawa 992, Japan. (Received August 1, 1975) ABSTRACT: The development of plastic deformation around the crack tip of known ductile polymers (polycarbonate and nylon 6) has been studied in connection with the Dugdale model of ductile yielding. It is shown that there are particular threshold stresses below which the yielding ahead of the crack tip would not be visible and that these are closely associated with the stable crack initiation. The shape of the plastic­ zone shows, in general, similar characteristics to those found in the Dugdale type yield­ ing, but the size and displacement do not compare well with the Dugdale predictions. A modified Dugdale model, where the cohesive stress varies linearly with the distance from the crack tip, is then successfully applied to the analysis of the experimental data. KEY WORDS Ductile Polymer I Polycarbonate I Nylon I Yielding I Dugdale Model I Plastic-Zone I Mises Criterion 1 Stable Crack I Crack-Opening Displacement I Since the pioneering work by Berry/ who be of practical importance to make an examina­ applied the Griffith theory to the fracture be­ tion of the applicability of linear fracture me­ havior of brittle polymers, there have been chanics to fracture behaviors of such ductile many studies to determine a fracture criterion polymers. From this point, for example, for them. Though nearly all of these studies Vincene investigated the stable crack growth are essentially based on a thermodynamic ap­ in poly(ethylene terepthalate) film to examine proach, a somewhat different approach, called the limitations of linear fracture mechanics. In fracture mechanics, was proposed by Irwin2 in this study, he concluded that linear fracture 1957. The concept of fracture mechanics itself, mechanics could not be applied to this class of although mainly developed for metals, is in­ ductile polymers. Following Vincent's work, dependent of the material to which it is applied. Fergunson and Williams8 have shown that Recent investigations have shown, indeed, that Vincent's data could be correlated with a con­ this concept is quite successfully applicable to stant crack-opening displacement criterion by both the brittle crack initiation and propagation using the Dugdale modeV which assumes non­ problems in different types of polymer fracture, linear elasticity of the material. such as static and dynamic fractures, 3 fatigue, 4 The energy balance concept has been con­ stress cracking, 5 and crazing. 6 sidered one of the most important methods for For the linearly elastic materials, or an characterizing a fracture criterion for ductile as elastic-plastic material where the plastic de­ well as brittle materials. For a conservative formation is small enough in comparison with system in ductile materials, one quite clearly the crack size, the fracture criterion obtained would expect that the work put into the speci­ from fracture mechanics is the same form as men is dissipated to create plastic deformations that of the Griffith theory. In many practical ahead of the crack tip. It is well known that cases, however, the majority of polymers are this dissipation is frequently several orders of ductile. These polymers usually exhibit con­ magnitude larger than the surface energy newly siderably greater amounts of nonlinear behavior created due to a propagating crack. In order prior to unstable fracture. Therefore, it can to evaluate separately the energy for plastic 181 I. NARISAWA, M. ISHIKAWA, and H. OGAWA deformations at the crack tip, one should cal­ culate the total amounts of the displacement cr : Applied stress and the stress distribution within the plastic cr 0 : Cohesive stress region. Several attempts have been made to i C Crack length obtain these quantities experimentally in semi­ s : Plastic zone ductile metals; however, the success is not very length a= C + s notable so far. If the Dugdale model can be applied to such ductile polymers, the energy balance considerations would be greatly simpli­ fied. Moreover, analytical comparisons can be made between various fracture criteria and a semi-empirical criterion, such as the crack­ opening displacement criterion, can gain a physical basis. Thus the Dugdale model pro­ lo=O----a vides the first stage of an approach to nonlinear fracture mechanics. Crack-opening In these circumstances, except for the few displacement limited attempts of Brinson10 and Bergkvist, 11 cr very little effort has been expended on polymers. Brinson has shown that the plastic region in Figure 1. The Dugdale model. polycarbonate can be identified by the photo­ elastic fringe shape, and he concluded that the the plastic-zone size have been studied. Of plastic-zone size was predicted by the Dugdale these, for its similicity, the Dugdale model is model. However, it should be noted that his the most practical for a computation of the method cannot always be useful for determining length of thin plastic-zones. As shown in Figure the plastic-zone sizes of other polymers, because 1, the assumptions of the Dugdale model for they are generally less sensitive to photoelasticity solutions are: than is polycarbonate. Bergkvist concluded, 1. An ideal elastic-plastic material (no stress conversely, that the plastic-zone size in poly­ or strain hardening effects) which flows after (vinyl chloride), which was measured from dis­ yielding at a constant uniaxial stress a0 • placements of an engraved grid on the specimen 2. Two separate load systems are applied on surface, could not compare well with the a cracked tensile sheet. One is the gross stress Dugdale predictions. This type of investigation a at infinity and another is a cohesive stress has just been started so that, in this field, it is a 0 on each part of the crack surfaces which of importance to increase the experimental data simulates a plastic-zone, which is a thin exten­ at the present time, as both Brinson and Berkvist sion of the crack lines. have emphasized at the end of their reports. 3. The plastic-zone length s is such that there As the first part of the series of papers which is no stress singularity at the crack tip. have been undertaken to elucidate the fracture The solution for the length of the plastic-zone mechanism in ductile polymers, the object of which results from the above assumptions can this paper is to measure the plastic-zone sizes be written. and shapes, and to compare these quantities with one of the most useful, existing nonlinear -=s 2 Sill. 2 ( --7r(J ) ( 1 ) a 2a predictions. 0 The estimation of the crack-opening displace­ NONLINEAR FRACTURE MECHANICS ment ¢> was obtained by Goodier and Field.12 (THE DUGDALE MODEL) This can be written Several elastic-plastic solutions with ana­ '1'---r!.- SCao 1n { sec ( --na )} ( 2) lytical and numerical methods for estimating nE 2a0 182 Polymer J., Vol. 8, No. 2, 1976 Fracture Processes in Ductile Polymers. I. where E is the Young's modulus of the material. For small aja0 these expressions are simplified to ( 3) ( 4) where K is the stress intensity factor. EXPERIMENTAL The specimens were cut along the machine direction from extruded polycarbonate film 39 f-l thick and nylon 6 film 41 f-l thick as rectangular Figure 3. Testing apparatus. strips, 20 mm wide and 150 mm long. All speci­ mens were used without annealing so that they sharp crack of controlled length was introduced were slightly anisotropic. Figure 2 shows their by driving a new razer blade while it was held stress-strain characteristics. Yield stresses used under a slight strain. The top radius of a crack for the calculation in the following experiments induced in this way was found to be not larger and obtained from the stress-maximum for poly­ than 1 f-l and there was no evidence of residual carbonate and the tangential intersection for stress at the crack tip. nylon, were 5.5 and 2.0 kg/mm2 , respectively. As shown in Figure 3, the tensile apparatus In the center of one edge of the specimen, a is suitable for using a polarizing microscope. The microscope can be moved in the two orthogonal directions and the distance of the movements can be read by use of dial-gauges within 0.1 f-l· The specimen was mounted hori­ 6 5.5 zontally between two clamps. To one of the clamps was attached a tensile load-cell (Toyo­ Baldwin, UK-2K-120) and the output was re­ 5 POLYCARBONATE corded on a Model SS-505D recorder (Toyo­ Baldwin). The specimens were extended at a strain of this stress and the plastic-zone sizes :E :>::..... were measured by use of both the dial-gauges ::::;'-" and the microscope. All measurements were made in air at 20°C and 65% RH. w ,..... U)""' RESULTS 2 Plastic-Zone Pro files The successive steps of the plastic deformation 1 occurring ahead of a crack tip in nylon and polycarbonate are shown in Figure 4. In both cases, as may be seen from these microphoto­ 0 graphs, the shapes of the plastic-zone show 0 5 10 wedge-type characteristics, and these shapes be­ ELONGA TJ ON (%) come clearly visible when the slow cracking at Figure 2. Stress-strain curves at 20°C (1.08%/min) the crack tip initiates. Alternatively, there for specimens of polycarbonate and nylon films. seems to be a threshold stress below which The values denote yield stresses. neither plastic deformation nor stable cracking Polymer J., Vol. 8, No. 2, 1976 183 I. NARISAWA, M. ISHIKAWA, and H.

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