Stress intensity factor solutions

Various approaches have been developed to obtain analytical and numerical expressions for the stress intensity factor associated with a wide variety of crack and loading geometries. These solutions are useful not only in developing fracture toughness testing techniques, but also in understanding the interaction of cracks with structure at all scale levels. 

1 Stress intensity factor solutions and test geometries 

It is useful to consider some stress intensity factor solutions for basic situations, especially those that relate to typical testing geometries used for fracture tough- 

The geometry given in Fig. 8.3, a crack loaded in uniaxial tension (mode I), has been discussed several times in this chapter. It is also shown in Fig. 8.16(a), emphasizing the presence of infinite boundaries (small crack size). For this geometry 

Cracks are often found at the outside surface of a material. For mode I loading (Fig. 8.16(b)), the solution is expected to be similar to that given in Eq. (8.25). Indeed, it is found that 

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Wu X.R. and Carlsson A.J. (1991). Weight Functions and Stress Intensity Factor Solutions. Pergamon Press, Oxford. 

Equation (3.42) may be used as a Green s function to develop stress intensity factor solutions for other loading conditions. 

Some commonly used specimens and stress intensity factor solutions are given in ASTM Method E-399 on fracture toughness testing [2]. Stress intensity factor solutions for other geometries are given in handbooks by Sih et al. [4] and Tada et al. 

If the stress intensity factor solution for the cracked component has the form of Eq. 9 and it is subjected to constant amplitude loading, then Eq. 15 becomes... 

A number of stress intensity factor solutions have been developed over the years. Several solutions are given in the review article by Hutchinson and Suo [ 19]. One example is the stress intensity factors for a bimaterial double cantilever beam subjected to uniform bending (Fig. 8). In this case... 

Linear Elastic Fracture Mechanics. Compendium of Stress Intensity Factors Solutions... 

C.B. Buchelet and WH. Bamford, Stress intensity factor solutions for continuous surface flaws in reactor pressure vessels, in Mechanics of Crack Growth ASTM STP S90, American Society for Testing and Materials, 1976, pp. 385 402. 

Mechanical Behaviour of Plastics Solution The stress intensity factor for this configuration is... 

Using the MBL formulation, we performed additional transient hydrogen transport calculations with L — 5.10, 9.96, 16.04, 21.36. 31.28. 41.63, 50.38 mm, stress intensity factor K, =34.12 MPaVm. T Icsa =-0.316, and zero hydrogen concentration C, prescribed on the outer boundary. For these domain sizes, we found the values of the effective time to steady state r to be 240. 608. 1105. 1538. 2297, 2976. and 3450 sec, respectively. Although the MBL approach does not predict the effective time to steady state accurately in comparison to the full-field solution, it can be used to provide a rough approximation. The non-dimensional effective times to steady-state r = Dl jb and the... 

Available theoretical solutions in dynamic fracture are few, and limited to finite or semi-infinite cracks in an infinite solid for Mode I, self-similar crack extension. Despite the above limitations, short of conducting detailed numerical analysis of the crack tip state of stress, these solutions must be used to deduce the characteristics of the crack tip state of stress, as well as to extract the dynamic stress intensity factor for elastodynamic fracture mechanics. In the following sections, a brief description of available theoretical solutions is presented. 

The model is formulated in terms of an integral equation which is solved with the condition at the boundary of the open crack and the bridging zone (.x — 0) that 6(0) = 5C. It is interesting to note that the structure of the rate-dependent problem is such that, aside from the material parameters, the solution is completely determined for a given crack velocity. For a given velocity, the value of the applied stress intensity factor, K, and the length of the cohesive zone, L, that maintains this condition is determined. Selected results are presented below. 

First, the elastic stress distributions of the un-notched specimens are obtained from a finite element analysis. For the PI un-notched specimen, the discrepancy between the finite element and the analytical result is very small (about 0.01%), thus validating the finite element calculation in terms of accuracy through the meshing and the type of element used. Therefore a similar calculation is conducted on the G1 un-notched specimen where the span to height ratio is smaller. The mismatch on the maximum stresses at the bottom and at the top of the beam between the finite element calculations and the analytical solution is 0.74% in tension and 0.79% in compression (and remains constant upon further mesh refinement). This estimation of the stress distribution is then used for the following evaluation of the stress intensity factor. 

ESC testing has been conducted at 50°C only, since at 70°C the detergent was immiscible in water even at very low concentrations. Fig. 8 shows crack initiation times obtained in a 10% detergent/water solution at varying applied stress intensity factor for the two materials examined (filled points). For comparison purposes, data obtained in air at the same temperature are also reported (empty points). For sufficiently high values of the applied stress intensity factor, a linear trend which is quite close to that obtained in the non-aggressive environment is observed. 

Testing was conducted at 50°C on HDPE-1 with different concentrations of the detergent in the water solution for an applied stress intensity factor of 0.35 MPa m. Initiation times and crack speeds obtained are shown in Figs. 11 and 12 respectively. Results show that, at least for the stress intensity factor value applied, the more pronounced effect of the ESC agent occurs for a detergent concentration of about 40% by volume. 

A particularly interesting feature of solutions like that given above is their interpretation as providing a source of screening for the crack tip. The use of the word screening in this context refers to the fact that the local stress intensity factor is modified by the presence of the dislocation and can result in a net reduction in the local stresses in the vicinity of the crack tip. In particular, the total stress intensity factor is a sum of the form... 

It is seen that, under this assumption, the stress function Z(z) satisfies the boundary conditions and is a solution to the problem. The assumption of A = 0, however, needs to be verified further through a consideration of the displacements (see Eqn. (3.27)). For the moment, the assumption is deemed to be correct, and the process for obtaining the stress intensity factor is considered. 

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