New Methods for Measuring Fracture Toughness

Other groups studying materials other than metals followed on from Griffith in defining toughness in terms of the energy. Thus for rubber, Rivlin and Thomas [4] used T (for tearing energy) in place of Gc in a nonlinear elastic analysis while Gurney, Atkins and Mai [5] used R for a range of materials. The elastic–plastic fracture parameter Jc is also, of course, an energy per unit area parameter. Here I will use fracture toughness to mean the energy per unit area to create a fracture surface and use the symbol Gc whatever the material or the test method. Most of the test methods have been aimed at characterising the toughness of structural materials. The first standards were designed for materials which could be regarded as obeying elastic fracture mechanics. That is to say that the specimens behaved elastically since the plastic deformation was sufficiently local that it did not affect the overall elastic response. This is the case for the ASTM K1c standards for metals [6] in which the specimens were assumed to be linearly elastic and obeyed linear elastic fracture mechanics (LEFM) rules. Interestingly the tests developed for rubber were based on non- linear elasticity and were analysed via elastic fracture methods. The latter determined T (Gc) and not Kc. Standards developed for polymers at ASTM and ISO [7] followed the metals scheme but determined Gc in addition to Kc. This was not considered necessary for metals since E was known and Kc could easily be converted to Gc. For polymers this was not so and the LEFM method was extended to include displacement measurements, and hence energy, to find both Gc and E. Standards have also been established for composite laminates and for adhesive joints which are LEFM based [8, 9]. Here the fibre reinforcement and the substrates provide the linear elastic response of the system. It is interesting to note that the behaviour of the specimens employed has a strong influence on the nature of the test. Griffith used cracked bars and beams and in most cases these are unstable when the crack initiates. Thus the beam and compact tension specimens used in Kc tests describe the crack initiation condition. This should not present any problems in theory since initiation is followed immediately by instability and a sudden load drop. In practice, however, plastic work and slow crack growth occur which give initially stable crack growth and arbitrary limits are set on degrees of nonlinearity in the loading curve to define initiation. Thus there is an arbitrary element in the definition of toughness in these tests which detracts from the value of the data obtained. There is also the issue of crack resistance curve behaviour. This is the ‘R’ curve effect where materials are supposed to have a fracture toughness which varies with crack length. Again this would obscure the initiation condition. To overcome this problem many tests are designed to give stable crack growth as in the double cantilever and tapered cantilever beam specimens used for composites and adhesives. Such tests are attractive because they enable the ‘R’ curve to be defined without resorting to arbitrary definitions. If initiation values are required they may be added to the ‘R’ curve. Such a situation exists for composites where the standard defines the ‘R’ curve with three definitions of initiation. Such stable crack growth tests are also widely used for environmental and fatigue testing of composites and adhesives. All these elastic based methods are highly successful in measuring toughness as is reflected in the several standards which exist. When the elastic dominance is lost, however, the situation is much less satisfactory. High toughness metals, which are important in many critical applications, result in substantial plastic deformation and associated crack blunting, prior to fracture. This led to the development of the J integral method [10]. The derivation of J is similar to that for G but for plastic deformation and is thus only valid for none unloading cases. However very useful schemes for measuring ‘R’ curves were developed and in which the crack growth was measured. Such methods sought to define an initiation value which would be identical to an elastically determined value. Subsequently this quest was abandoned because of the difficulty of separating crack growth from crack blunting, particularly for low yield stress materials. More recent Jc standards define an ‘R’ curve and a J at an arbitrary crack growth [10]. A similar situation pertains for polymers and other soft materials where the blunting versus crack growth problem is acute. It is also generally accepted that that ‘R’ curves derived from these methods are geometry, and size, dependent and thus not true material properties. However such test data can be of value for comparative testing and for many materials is all that is available. An alternative, which is usually applied to thin sheet materials, is the Essential Work of Fracture [11, 12]. Here it is assumed that the total energy dissipated may be partitioned into the fracture (essential) work and other energy dissipated. The essential work is equated to Gc or Jc and the method has proved very useful for thin section materials. For thick sections however, it has been found difficult to apply because the extrapolation is no longer linear. The method does use a scheme which has proved useful in other tests in that specimens of different ligament sizes are tested and the essential work is found from extrapolating to a zero ligament. This lack of effective methods for testing soft and tough materials is a limitation on the use of fracture mechanics methods in several fields. It is a limitation in evaluating engineering materials such as polymers used in pipe systems and also in polymers which are modified to give high toughness with low yield stresses. There is also a need to evaluate biological materials. The toughness of foods correlates with texture as perceived in chewing and toughness characterisation is an attractive alternative to more common subjective methods such as tasting panels. The resistance of flesh to cutting and penetration is important in medical applications. Such materials have soft, rubber like, stress–strain behaviour with a wide range of toughness. Recent studies have shown that cutting and machining tests are a possible solution to these problems. The analysis of machining goes back over 100 years and has concentrated on calculating the plastic work done in forming chips when removing layers of material. The inclusion of the fracture energy was much debated in the early years but it was concluded that the contribution was negligible compared to that from plasticity and friction. It was pointed out by Atkins in the 1990s [13, 14] that this was erroneous since it was based on using the surface work not including the local plasticity (or damage) as Irwin had demonstrated to be necessary for most materials. The basic idea is that layers of varying thickness may be removed by orthogonal cutting by using a sharp tool with a low rake angle (i.e. a steep tool face).This induces shear yielding in the chip and plastic work proportional to the chip thickness. During each test the cutting force and the transverse force need to be measured and, for a fixed cutting speed, these will be constant since a steady state is established. These two forces enables the data to be corrected for friction energy and when the corrected cutting force is plotted versus the thickness removed then a straight line results with a positive intercept of Gc. This method has been shown to work well for a range of polymers [15] and particularly so for those with low yield stresses. Comparisons for materials where LEFM tests can be applied have been encouraging. The method also works well for metals and data is available on steel, brass and aluminium alloy [16]. A closely related method is wire cutting which is well suited to very soft materials [17]. Here the force is measured to cut the sample for wires of a range of diameters. The plastic or damage energy is proportional to the diameter of the wire so that linearity ensues and there is a positive intercept of Gc. This scheme has worked well for foods and particularly cheese where, of course, the basic idea comes from. It is also successful in cutting gels where Gc as low as 1 J m−2 can be measured [18]. These methods overcome the crack blunting problem by using a sharp tool and a flat fracture is produced. They are steady state tests and can determine toughness at a range of speeds determined by the rate of cutting and in general they are stable. The corrections for friction and plasticity energy are significant but careful experimental work has shown that toughness values with variations of 20% can be obtained. It is early days in their application but they are certainly very promising for evaluating difficult materials.