Plane Strain and the Brittleness of Plastics

8 Plane Strain and the Brittleness of Plastics

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on September 11, 2016 | http://pubs.acs.org Publication Date: June 1, 1976 | doi: 10.1021/ba-1976-0154.ch008

A. F. YEE, W. V. OLSZEWSKI, and S. MILLER Synthesis and Characterization Branch, Chemical Laboratory, General Electric Corporate Research and Development Center, Schenectady, N. Y.

Double-grooved specimens were used to study the failure of PC, PC/PE, PET, ABS, and HIPS during transitions from plane stress to plane strain. The yield behavior of PC is consistent with a von Mises-type yield criterion; plane strain reduces its elongation. The yield behavior of PC/PE is consistent with a Tresca-type yield criterion; plane strain appears to be relieved by voiding around the PE particles. PET undergoes a ductile-to-brittle transition; its behavior is consistent with a von Mises-type yield locus intersected by a craze locus. The yield behavior of ABS and HIPS is not significantly affected by the plane-stress-to-plane-strain transition. Plane strain alone does not necessarily cause brittleness.

V V T h e n notched polymer specimens are subjected to rapid tension or flexure, brittle failure frequently occurs. Whether or not a notched specimen fails brittlely at a given temperature depends on the sharpness of the notch, the thickness of the specimen, and the speed of deformation. Sharpness, thickness, and speed are all relative and interdependent terms. The particular combination of these three variables that causes a ductileto-brittle transition depends on the polymer and the thermal-mechanical history of the specimen being tested. For example, the notched Izod impact strength of polycarbonate has been investigated by systematically changing only the following variables: notch sharpness (I), specimen thickness (1,2), speed ( 3 ) , degree of annealing (1,4), etc. In each case a ductile-to-brittle transition has been observed. In the notched Izod impact test, with the exception of some rubber modified plastics, the fracture surface of a specimen that has failed ductilely exhibits significant inward collapsing on the sides of the impact bar near the notch, indicating that a large amount of plastic flow has 97 Deanin and Crugnola; Toughness and Brittleness of Plastics Advances in Chemistry; American Chemical Society: Washington, DC, 1976.


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Figure 1. Typical appearances of failed notched Izod impact specimens

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taken place. In contrast, the fracture surface of a specimen that has failed brittlely has sides that remain straight and is relatively flat and featureless on a macroscopic scale (Figure 1). Therefore, the ductile-tobrittle transition is thought to be caused by a plane-stress-to-plane-strain transition (5). The stresses near the root of a notch are extremely complex; and the stress analysis becomes exceedingly difficult when the strain is large, as is the case when yield or failure is imminent. A sharp notch causes constraints and introduces a state of triaxial tension behind the root of the notch (5). This state of stress is consistent with LeGrands observation of the growth of a flaw behind a notch in a bar of polycarbonate (4). A blunt notch causes constraints when the thickness of the specimen is large. Such a notch can also introduce a state of triaxial tension. While it is desirable to investigate the behavior of polymers in a well-defined state of triaxial tension, it is difficult to accomplish experimentally. However, as we demonstate below, a state of plane strain is relatively easy to produce. The relationship between plane strain and brittleness of plastics is the subject of our investigation. Consider a body in the Cartesian coordinates x x , and * . Plane strain is defined as the state the body is in when the displacement vector /xi vanishes and when the orthogonal displacement vector components /x and /x are functions of x and x only (6). Although not rigorously correct, we will substitute strain c for displacement /x in the above definition to simplify the following discussion. The yield criteria of polymers have been reviewed by Ward (7) and more recently by Raghava et al. (8). Except for the craze yield criteria of Sternstein and Ongchin (9) and Bowden and Oxborough (10), most of the yield data can be described by a pressure-modified, von Mises-yield criterion. The corresponding yield surface is everywhere convex. A typical yield locus on the o-i-o- plane is shown in Figure 2. If the assumpu

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Deanin and Crugnola; Toughness and Brittleness of Plastics Advances in Chemistry; American Chemical Society: Washington, DC, 1976.

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tion of the normality rule is correct for polymers (11, 12), then the strain vector c is normal to the yield locus. [Strain is a tensor; however, for convenience it will be treated here as a vector.] The plane strain point of ci = 0 on this locus can then be located by drawing a tangent to the locus such that it is perpendicular to the 0. For a hypothetical, two-dimensional material, the stress states in this region could lead to voiding. For a real three dimensional material, a volume dilatation takes place when ci + c + 3 > 0, thus c > — €2 must be satisfied for dilatation. This can come about as a result of the contraction ratio being smaller than Vz or as a result of geometric constraints or rigid particle inclusions. [Contraction ratio is the ratio of lateral contraction to axial elongation in a tensile test.] For a viscoelastic material, one other effect that must be considered is the time dependence of Poisson's ratio (13). For example, lateral contraction can lag behind longitudinal stretching when such stretching takes place rapidly, thus producing time dependent dilatation. Dilatation presumably causes voiding and finally crazing (14, 15) and brittle fracture.


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However, not being able to independently control c , our investigation is limited only to observations of yield behavior made during a controlled plane-stress-to-plane-strain transition. 3


Specimen Tensile plane strain can be created by pressurizing a fixed-end thinwalled tube or, more simply, by imposing elastic constraint on the edges of a wide strip tensile specimen. Corrigan et al. used a specimen with grooves machined on both sides of a weldment to test the biaxial strength of welds in steel (16). Whitney and Andrews tested PMMA in tensile plane strain by clamping a wide thin sheet in wide grips (17). However, the distance between the two grips was so large that it is doubtful that plane strain was obtained. Recently, Lee tested wide HIPS strip specimens with grooves at oblique angles to the tensile axis to vary the biaxiality, including plane strain (18). The stress distribution in a biaxial strip test for polymeric materials has been theoretically analyzed by Cost and Parr (19). Their results can be used to aid in the design of the specimen geometry and the interpretation of the results. The specimen we used is illustrated in Figure 3. The ideal specimen profile dimensions have been found to be T 3t and b— St to 5t (16,

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Shape of the specimens. Notations for the dimen sions and coordinates are also shown.


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0 — - — 0 DISTANCE FROM EDGE

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Figure 4. Schematic of the transverse strain distribution and the per cent of specimen width not in plane strain (e = 0) t

18, 20). Elastic constraint is effected when the strip width W is large compared with b, but non-zero transverse strains exist (i.e., ci ^ 0) at the groove edges regardless of W. If the specimen is narrow, these finite transverse strain zones overlap and the stress in the specimen is nearly uniaxial. If the specimen is wide, plane strain (ci = 0) is realized in a substantial portion of the specimen. This situation is schematically illustrated in Figure 4. Cost and Parr have shown that W/b has to be five in order that 80% of the specimen will be within 10% of the axial stress that an ideal infinitely wide specimen would have. They have also shown that differences in Poissons ratio can be accounted for by a simple scaling factor (19). Since most rigid polymers have Poissons ratio between 0.3 and 0.4, we expect substantial plane strain when W/b > 5. In these experiments b was fixed in all cases. However the thickness t was more difficult to control because of machining accuracy limitations. In plotting the data, smaller scatter was obtained by plotting yield stress against W/t. The tensile yield stress variation as a function of W for a material which has a von Mises-type yield locus is illustrated schematically in Figure 5. This variation is caused by the fact that as the width of the specimen increases, the biaxiality also increases toward the asymptotic value at plane strain. If the material obeys the von Mises yield criterion exactly, the plane strain yield stress should be 15% higher than it would be for simple tension. On the other hand, if the material obeys the Tresca yield criterion, the plane strain yield stress should be identical


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w/t Figure 5. Schematic of yield stress (in the x direction) vs. W/t for a von Mises-type polymer 2

to that of simple tension, and no variation in yield stress with respect to W is to be expected. At large W/b, the stress path, i.e., the ratio of o-i/

Plane Strain and the Brittleness of Plastics

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