Crack, penny-shaped

There is considerable literature on material imperfections and their relation to the failure process. Typically, these theories are material dependent flaws are idealized as penny-shaped cracks, spherical pores, or other regular geometries, and their distribution in size, orientation, and spatial extent is specified. The tensile stress at which fracture initiates at a flaw depends on material properties and geometry of the flaw, and scales with the size of the flaw (Carroll and Holt, 1972a, b Curran et al., 1977 Davison et al., 1977). In thermally activated fracture processes, one or more specific mechanisms are considered, and the fracture activation rate at a specified tensile-stress level follows from the stress dependence of the Boltzmann factor (Zlatin and Ioffe, 1973). 

Stresses can can be concentrated by various mechanisms. Perhaps the most simple of these is the one used by Zener (1946) to explain the grain size dependence of the yield stresses of polycrystals. This is the case of the shear crack which was studied by Inglis (1913). Consider a penny-shaped plane region in an elastic material of diameter, D, on which slip occurs freely and which has a radius of curvature, p at its edge. Then the shear stress concentration factor at its edge will be = (D/p)1/2.The shear stress needed to cause plastic shear is given by a proportionality constant, a times the elastic shear modulus,... 

Figure 16. SEM images of the fracture surface of an RH-cycled (80-120% RH) sample, showing crack initiation and growth from a craze site near the anode and membrane interface. A crazing site found on Hi-RH cycled samples. The half-penny shaped craze closely resembles crazes formed on free surfaces.

In the above discussion it is assumed that the volume displaced from the cavity by diffusion is the sole contribution to the creep rate. However, an additional elastic displacement associated with cavities also contributes to the creep rate. This form of cavitation creep is important in fibrous composites, in which crack-like cavities propagate along the fiber interface. Termed elastic creep, this type of creep was first analyzed by Venkateswaran and Hasselman112 and later by Suresh and Brickenbrough.113 As with other forms of cavitation, both the nucleation rate and the growth rate are important. From Venkateswaran and Hasselman, the creep rate, e, for a body containing penny-shaped cavities distributed throughout the volume is... 

Fig. 1. Penny shape craze grown from a defect. Note the small defect and the craze (3 bright fringes) without crack ...

Fig. 9 Diagram showing the variation of "equivalent normal stress" in PSZ phase with the orientation angle of penny-shaped crack (0) and the stress triaxiality (jS).

Figure 4.4. Schematic diagram of a circular (penny-shaped) crack inside a large body, snb-jected to nniformly applied, remote tensile stress perpendicnlar to the crack plane.

For this penny-shaped crack model specimen, the crack diameter would be the only dimension of concern the other dimensions would be taken to be very large. The crack size requirement may be considered by writing the effective crack diameter 2a in terms of the plane strain fracture toughness Kic and the yield strength ys of the material by using Eqn. (4.7) i.e., for the conceptual case where yielding and fracture occur concurrently. 

It is clear, so far, that the crack diameter is the characteristic dimension of the simple specimen (with a penny-shaped crack in an infinitely large body) under discussion. Based on cases I and II, there should be a useful lower limit for lUo = A Kjc/(Tys), where A > 1.5. This useful lower limit cannot be deduced theoretically at present because of the lack of a detailed understanding of the processes of fracture and the inability to model the deformation of real materials. It must be established experimentally through large numbers of Kjc tests, covering a representative range of materials. 

The aforementioned specimen with a penny-shaped crack and similar specimens facilitate simple and straightforward measurements of Ki, at least in principle. They are impractical, however, for a number of good reasons. These necessarily large specimens are inefficient with respect to the amount of material and the loading capacity of the testing machine that would be required. They may not reflect the actual microstructure and property of the size of material of interest, and cannot discern directional properties of the materials. 

The pop-in concept was first developed by Boyle, Sulhvan, and Krafft [3] and forms the basis of the current Kic test method that is embodied in ASTM Test Method E-399 [2]. The basic concept is based on having material of sufficient thickness so that the developing plane stress plastic zone at the surface would not reheve the plane strain constraint in the midthickness region of the crack front at the onset of crack growth (see Fig. 4.2). It flowed logically from the case of the penny-shaped crack, as shown in Fig. 4.4 in the previous subsection. 

A specimen of finite thickness may be viewed simply as a shce taken from the penny-shaped crack specimen (Fig. 4.6). As a penny-shaped crack embedded in a large body, the crack-tip stress field is not affected by the external boundary surfaces and plane strain conditions that prevail along the entire crack front. As a slice, however, the crack in this alternate specimen is now in contact with two free... 

Figure 4.6. A finite-thickness specimen sliced from a large body that contains a penny-shaped crack.

If the specimen is very thick i.e., with thickness B much greater than the plastic zone size, or Kic/oysY), the constraint condition along the crack front in the midthickness region is that of plane strain and is barely affected by plastic deformation near the surfaces. Abrupt fracture crack growth) will occur when the crack-tip stress intensity factor reaches the plane strain fracture toughness Kjc. The load-displacement record, similar to that of the penny-shaped crack, is depicted by Fig. 4.7a. 

Fracture Model. A powerful fracture model based on Statistical Crack Mechanics (SCM) is being developed at Los Alamos ((>). In this model, the rock is treated as an elastic material containing a distribution of penny-shaped flaws and cracks of various sizes and orientations. Plasticity near crack tips is taken into account through its effect on the fracture toughness. 

The median crack is a single, penny-shaped crack nucleated beneath the apex of the plastic zone created by the indentor. The diameter of the median crack is comparable with the indent size, and a median crack is not visible in a polycrystalline, opaque material. The driving force for nucleation of the median crack is the elastic tensile stress developed normal to the indentation direction at the elastic-plastic boundary when the external load is relaxed. Nucleation of a median crack depends on the presence of a suitable flaw. Once nucleated, the median crack will propagate spontaneously to a stable flaw size. The critical flaw size for growth is ... 

The link between idealized microcracks and modulus has been established by multiple researchers For isotropically distributed penny-shaped microcracks, Hassehnan and Singh showed that the functional form for the relationship between isotropic crack density n, elastic modulus E of the microcracked material, and the uncracked modulus E is... 

Based on the micrograph observation in Fig. 7.10, AE wave due to debonding of stainless steel from base metal is simulated. As illustrated in the figure, a penny-shaped tensile crack of 10 pm diameter and 1 pm crack opening is assumed from the observation. Results of detected AE wave... 

Fig. 7.11. (a) Detected AE waveform and (b) synthesized waveform due to a penny-shaped tensile crack of debonding. 

The blister test has also been used to study mould adhesion. It requires a special mould, but could probably be used with any moulding machine. It is well suited to measuring the low levels of adhesion involved in mould release. Essentially a disc, like a beer mat, is moulded and the pressure is applied to displace it through a small hole ( penny-shaped crack ) at its centre. The detachment pressure can be related to the fracture energy. 

Collins, W.D. (1962) Some axially symmetric stress distributions in elastic solids containing penny-shaped cracks. III. A crack in a circular beam. Proc. Edinburgh Math. Soc., 13(2) 69-78. 

In relation (4.27), Cq refers to the penny-shaped flaw size at the moment of crack initiation. Kq is an intensity factor chosen as Kq = 2 MPa.m. The bridging-zone length, Db, prior to critical fracture, has a bridging stress, p, assumed to be constant. For the derivation of Eq. (4.27), the reader should consult the work of Chien-Wei Li et al. [34]. The estimated p for the FG, medium and... 

Failure criteria of the form of (18) have been considered by others. Indeed, equation (4c) of Ref. 3 is a special case of (18) with Q - l/(l-0.5i ), where v is Poisson s ratio. This failure criterion comes from the assumption that fracture occurs when the local (tensile) stress at some point on the surface of the crack cavity reaches the ultimate strength of the material. Equation (4c) of Ref. 4 derives from results of Paul and Mirandy (Ref. 9) for a penny-shaped crack. 

Smith, F.W., Alavi, M.J., 1985. Stress intensity factors for a penny shaped crack in a half plane. Eng. Fract. Mech. Sci. 20, 403. 

Fig. 7. Fracture surface in polyacetal resin specimen subjected to tensile fatigue. The penny-shaped crack initiated at a void or other inhomc eniety at its center (25).

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