The effects of different boundary conditions on three-dimensional cracked discs under anti-plane loading

• Three-dimensional discs weakened by through-the-thickness cracks under Mode III are modelled.
• A synthesis of the magnitude of the stress state through the thickness of the disc is carried out.
• The variability of local SED through the thickness of the disc is analysed.
• Numerical analyses are compared with the trend according to the developed theoretical frame.
• The capability of the SED to capture the combined three-dimensional effects is discussed.

Accordingly to the recent multi-scale model proposed by Sih and Tang, different orders of stress singularities are related to different material dependent boundary conditions associated with the interaction between the V-notch tip and the material under the remotely applied loading conditions. This induces complex three-dimensional stress and displacement fields in the proximity of the notch tip, which are worthy of investigation.Starting from Sih and Tang's model, in the present contribution the authors propose some analytical expressions for the calculation of the strain energy density (SED) averaged over a control volume embracing the V-notch tip. The expressions vary as a function of the different boundary conditions. Dealing with the specific crack case, the results from the analytical frame are compared with those determined numerically under linear-elastic hypotheses, by applying different constraints to the through-the-thickness crack edges in three dimensional discs subjected to Mode III loading. Free–free and free–clamped cases are considered.Due to three-dimensional effects, the application of a nominal Mode III loading condition automatically provokes coupled Modes (I and II). Not only the intensity of the induced modes but also their degree of singularity depend on the applied conditions on the crack flanks.The variability of local SED through the thickness of the disc is investigated by numerical analyses and compared with the theoretical trend. The capability of the SED to capture the combined three-dimensional effects is discussed in detail showing that this parameter is particularly useful when the definition of the Stress Intensity Factors (SIFs) is ambiguous or the direct comparison between SIFs with odd dimensionalities is not possible.