PSI - Issue 42

Tamás Fekete et al. / Procedia Structural Integrity 42 (2022) 1467–1474 Tamás Fekete, Éva Feketéné-Szakos / Structural Integrity Procedia 00 (2019) 000–000

1474

8

1 cr pl ε –where gradual softening starts– E St has an inflection point, at 2 cr

pl ε an extremal

Fig. 2 clearly shows that at

point, while at 3 cr pl ε –at the onset of fast fracture– the singularity is degenerate, as indicated by the abrupt change – discontinuity– in E St . In Fig. 3, Σ –the flow curve– reaches its maximum at 1 cr pl ε . At 2 cr 3 cr pl ε the singularity is also degenerate for this case. Remember that Σ is the derivative of E H with respect to ε pl . Fig. 2 and 3 clearly show that as the crx pl ε points move towards each other –as shown separately in Fig. 4–, the originally regular singular points, 1 cr pl ε and 2 cr pl ε –describing smooth, soft transitions– become part of a strong, degenerate singularity, making it even more degenerated. The trends of the Σ curves in Fig. 3 correctly reflect the trends observed in the tensile tests of ageing SM s, showing that the approach is promising for further development. pl ε progressive softening starts, and at The paper summarized the theoretical framework of the NLFTFM -based methodology for SIC s of ageing SLL s. Then, the basics of an ageing model were presented. An example illustrated the basics of the theory. A specific feature of the ageing model is that –with a few additional considerations, compatible with the NLFTFM – it provides a qualitatively sound phenomenology of the ageing-related embrittlement of SM s. The novelty of the approach lies in the fact that, when describing the long-term behavior of a SM , particular attention is paid to the evolution of the system of singularities encoded in the description. The trends of simulations using the model correctly reflect those observed in tensile tests of ageing SM s. Though much remains to be done, the presented approach seems promising in solving challenging issues, such as the behavior of SM s in the ductile to brittle transition temperature regime. Arnold, V.I., 1984. Catastrophe Theory . Springer-Verlag, Berlin Heidelberg New York Tokyo. Bažant, Z.P., Cedolin, L., 1991. Stability of Structures –Elastic, Inelastic, Fracture and Damage Theories– . Oxford University Press, New York, Oxford. Béda, Gy., Kozák, I., Verhás, J., 1995. Continuum Mechanics . Akadémiai Kiadó, Budapest. Chen, X.H., Mai, Y.W., 2013. Fracture Mechanics of Electromagnetic Materials. Nonlinear Field Theory and Applications . Imperial College Press, London. Fekete, T., 2019. Towards a Generalized Methodology for Structural Integrity Calculations of Large-Scale Pressure Vessels, Procedia Structural Integrity 17 464–471. https://10.1016/j.prostr.2019.08.061 Fekete, T., 2022. The Fundaments of Structural Integrity of Large-Scale Pressure Systems, Procedia Structural Integrity 37 779–787. https://10.1016/j.prostr.2022.02.009 Feketéné Szakos, É. 2018. Az id ő skori tanulás támogatása IKT eszközökkel. (Support of Learning in Old Age with ICT Tools) In: Fodorné Tóth, K. (Ed.): Social and Economic Benefits of University Lifelong Learning: Research – Development and Innovation . ’MELLearN - Fels ő oktatási Hálózat az Életen át tartó tanulásért’ Egyesület, Debrecen University. Gilmore, R., 1993. Catastrophe Theory for Scientists and Engineers . Dover Publications, Inc., New York. Kaplan, M., Sanchez, M., Hoffman, J. 2017. Intergenerational Pathways to a Sustainable Society . Springer, Heidelberg. Malgrange, B. 1964, The preparation theorem for differentiable functions, In: Atiyah, M., 1964. (Ed.): Differential Analysis: Papers Presented at the Bombay Colloquium, 1964. Oxford University Press, London. 203–208. Maugin G.A., 2009. On inhomogeneity, growth, ageing and the dynamics of materials. J. of Mechanics of Materials and Structures , 4:4, 731–741. Ne’eman, Y., Hehl, F.W., Mielke, E.W., 1993. The Generalized Erlangen Program and Setting a Geometry for Four-Dimensional Conformal Fields , Report, DOE / ER/40200-323, Texas University, Austin, TX. http://inis.iaea.org/search/search.aspx?orig_q=RN:26027045 Oldofredi, A., Öttinger, H.C., 2021. The dissipative approach to quantum field theory: conceptual foundations and ontological implications, European Journal for Philosophy of Science , 11 18 https://doi.org/10.1007/s13194-020-00330-9 Öttinger, H.C., 2017. A philosophical approach to quantum field theory. Cambridge University Press, Cambridge, New York. Steinmann, P., 2022. Spatial and Material Forces in Nonlinear Continuum Mechanics. A Dissipation-Consistent Approach . In: Solid Mechanics and Its Applications 272. Springer Nature, Cham. https://doi.org/10.1007/978-3-030-89070-4 Stewart, I., 1981. Applications of catastrophe theory to the physical sciences, Physica D : Nonlinear Phenomena , 2 2 245–305. https://doi.org/ 10.1016/0167-2789(81)90012-9 Thom, R., 1975. Structural Stability and Morphogenesis , An Outline for a General Theory of Models . W. A. Benjamin Inc, Reding, MA. Wilczek, F., 2016. Entanglement Made Simple, Quantamagazine , April 28, 2016, https://www.quantamagazine.org/print. Yavari, A., Marsden, J.E., Ortiz, M., 2006. On spatial and material covariant balance laws in elasticity, Journal of Mathematical Physics 47 4 042903 https://doi.org/10.1063/1.2190827 Zeeman, E.C., 1979. Catastrophe Geometry in Physics: A Perspective, In: Gattinger, W., Eikemeier, H., (Eds.), Structural Stability in Physics, Proceedings of Two International Symposia on Applications of Catastrophe Theory and Topological Concepts in Physics, Tübingen, Fed. Rep. of Germany, May 2-6 and December 11-14, 1978 . Springer-Verlag, Berlin Heidelberg New York References 4. Summary, Conclusions