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This article introduces the Special Issue on Sustainable Management of Heritage Buildings in long-term perspective.

It starts by reviewing the gaps in knowledge and practice which led to the creation and implementation of the research project SyMBoL—Sustainable Management of Heritage Buildings in long-term perspective funded by the Norwegian Research Council over the 2018–2022 period. The SyMBoL project is the motivation at the base of this special issue.

The editorial paper briefly presents the main outcomes of SyMBoL. It then reviews the contributions to the Special Issue, focussing on the connection or differentiation with SyMBoL and on multidisciplinary findings that address some of the initial referred gaps.

The article shortly summarizes topics related to sustainable preservation of heritage buildings in time of reduced resources, energy crisis and impacts of natural hazards and global warming. Finally, it highlights future research directions targeted to overcome, or partially mitigate, the above-mentioned challenges, for example, taking advantage of no sestructive techniques interoperability, heritage building information modelling and digital twin models, and machine learning and risk assessment algorithms.

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. The paper aims to discuss these issues.

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 analysed 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.