Plane-strain condition

Substantial work on the appHcation of fracture mechanics techniques to plastics has occurred siace the 1970s (215—222). This is based on earlier work on inorganic glasses, which showed that failure stress is proportional to the square root of the energy required to create the new surfaces as a crack grows and iaversely with the square root of the crack size (223). For the use of linear elastic fracture mechanics ia plastics, certaia assumptioas must be met (224) (/) the material is linearly elastic (2) the flaws within the material are sharp and (J) plane strain conditions apply ia the crack froat regioa. 

The stress—strain relationship is used in conjunction with the rules for determining the stress and strain components with respect to some angle 9 relative to the fiber direction to obtain the stress—strain relationship for a lamina loaded under plane strain conditions where the fibers are at an angle 9 to the loading axis. When the material axes and loading axes are not coincident, then coupling between shear and extension occurs and... 

Note that the symbol Gic is used for the plane strain condition and since this represents the least value of toughness in the material, it is this value which is usually quoted. Table 2.2 gives values for G c for a range of plastics. 

Critical Stress Intensity Factor It has become common to use AT scc> the critical stress intensity factor, as a measure of the resistance of an alloy to s.c.c. Tests are performed on specimens which are precracked by a fatigue machine and must be of sufficient dimensions to ensure plane strain conditions. Recommendations on precracking and dimensions are given elsewhere . ... 

Fig. 8.52 Initial stress intensity factor and time to failure for a susceptible titanium alloy tested in a neutral aqueous environment under plane strain conditions...

This plastic deformation is localised around the crack tip and is present in all stressed engineering materials at normal temperatures. The shape and size of this plastic zone can be calculated using Westergaards analysis. The plastic zone has a characteristic butterfly shape (Fig. 8.83). There are two sizes of plastic zone. One is associated with plane stress conditions, e.g. thin sections of materials, and the other with plane strain conditions in thick sections-this zone is smaller than found under plane stress. 

There are a number of restrictions on the test for for it to be a valid measure of plane strain fracture toughness. Firstly, the plastic zone must not extend through the test piece and secondly the thickness of the material must be such that the test is conducted under plane strain conditions. 

For a rectangular rubber block, plane strain conditions were imposed in the width direction and the rubber was assumed to be an incompressible elastic solid obeying the simplest nonhnear constitutive relation (neo-Hookean). Hence, the elastic properties could be described by only one elastic constant, the shear modulus jx. The shear stress t 2 is then linearly related to the amount of shear y [1,2] ... 

We obtained full field solutions for an axial crack on the ID surface of a pipe under plane strain conditions. The pipe has an outside diameter of 40.64 cm and a wall thickness h = 9.52 mm. The depth of the axial crack is a = 1.9 mm (a/h = 0.2) (Fig. 2). In the absence of hydrogen, we assume a finite crack-tip opening displacement (CTOD), ha = 2 pm. 

K[c Critical stress intensity factor in mode I and plane strain conditions... 

For the four-point bend geometry used in the present experiments and under plane strain conditions, the C integral is given by 30... 

Since crazing represents a cavitational form of plasticity, it is clear that crazes play an important role in the fracture of polymers. Crazing is, generally, involved when PC fails under plane strain conditions, e.g. in fracture mechanics tests on thick samples with sharp notches where high triaxial stresses are built... 

There can be no doubt as to the importance of plane strain conditions for the fracture of plastics especially where sharp notches and thick sections are concerned. Such conditions nearly always lead to brittle or semi-brittle fracture. Vincent has shown that the notch sensitivity in a braod range of amorphous and crystalline polymers is increased as the testing temperature is lowered and the loading rate is increased. Before fracture occurs, amorphous plastics often craze under these conditions. The complex questions of craze initiation, propagation and transformation into a crack have been treated extensively for amorphous polymers in the first three chapters of this book (see also The problem becomes more complicated when... 

The table shows the force needed to fracture single-edge bars of several polymers at room temperature in a three-point bending test. Bar dimensions are IT = 10 mm, B = 6 mm, a = 5 mm the test span S is 80 mm. The second column in the table gives the corresponding yield stress. Calculate Kic for each polymer, and indicate in which cases valid plane strain conditions exist. 

And finally, it is worth recalling that Kjc and Gjc are two quantities which are related to each other, in plane strain conditions, by the well-known relationship [11] ... 

The fracture process is investigated for two glassy polymers polymethyl methacrylate (PMMA) and polycarbonate (PC) which are generally thou t to show a brittle and a ductile response respectively and thus selected to illustrate the method. These materials consist of commercial sheets (from Goodfellow) of 10 mm thickness which ensures plane strain conditions for both materials. Caution about plane strain conditions concerns primarily PC which is prone to develop plasticity and a 10 mm thickness appears reasonable according to analysis of the influence of the thickness on its toughness found in [6, 7]. 

Substitution of Eqs. (9) into Eqs. (8) and subsequent differentiation with respect to lead to the equilibrium equations in terms of microstresses and microstrains (i.e. strains averaged across the layer thickness). To exclude the latter, constitutive equations for the damaged layer and the outer sublaminate, equations of the global equilibrium of the laminate as well as generalised plane strain conditions are employed. Finally, a system of coupled second order non-homogeneous ordinary differential equations is obtained... 

Owing to the difficulty of moulding thick sheets, most fixture measurements on pdymers have been made on specimens less than 7 mm thick. This thickness is not sufficient to produce plane strain conditions in most tou or ductile polymers. The difference between plane stress and plane strain fracture has hitherto been widely regarded as unimportant in HIPS and other rubber-modified jdastics, since crazing is itself a plane-strain process, and little attempt has been made to work with thick specimens. However, recent work by Parvin and Williams has shown that there is a very marked transition between plane stress and plane strain fracture in and it is clear that this aspect of rubber-toughening will requite closer attention in the future. 

Figure 15 summarises the results of these experiments. Values of G/c are given for temperatures up to 10 °C, above wdiich LEFM methods do not apply, and values of R are given instead. Below the Tg of the mbber at about —90 °C, fracture is completely brittle, with no sign of stress-whitening, and a very low G/c- Between —90° and 10°, stress-whitening occurs at the tip of the notch, and Gjc increases with terrperature as the extent of the yield zone increases. It is dear from the work of Parvin and Wflliams that fracture takes place under plane-stress rather than plane-strain conditions in the stress-whit ed specimens. Away from the whitened zone. 

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