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

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3. Theoretical Framework

As mentioned above, SI is the discipline that seeks to determine –with scientific precision and using engineering procedures– the conditions under which, and the period to which, the SI of a system can be ensured. It is of particular importance for LSS s, as they are not replaceable neither as a whole, nor in their safety critical components. From the perspective of sustainability, if the operating technology is environmentally friendly, LSS s should be operated for the maximum SL of safe operation: a longer SL will significantly reduce the ecological footprint of the investment. On one hand, ageing processes in the system need to be slowed down and their progressive damage delayed as much as possible, through SH monitoring procedures and consequent mitigating actions. Moreover, the predictive power of SIC s determining the TAL of LSS s is to be increased, since current, standard-based SIC s bear the fingerprints of decades-old concepts of engineering mechanics. This means that most models in use are based on a reversible, mechanistic predominantly linear worldview. The introduction of a more consistent, irreversible worldview-based theoretical model having its roots in modern Thermodynamics, seems to be a major step forward – see Öttinger (2017), Oldofredi, Öttinger (2021), Fekete (2019, 2022), Steinmann (2022)–. Currently, the ‘ weakest link ’ in standard-based SIC methodologies developed for LSS s is the lack of a fundamental, holistic model supporting the coupled modelling of ageing and fracture . Some 90 years after Griffith's first model of Fracture Mechanics ( FM ) in 1921, Chen and Mai –using the Classic Thermodynamics of Irreversible Processes ( CTIP )– developed the first version of a Nonlinear Field Theory of Fracture Mechanics ( NLFTFM ) –see Chen, Mai (2013)–. NLFTFM seems promising for solving FM problems arising in dissipative settings, a highly significant issue, as dissipative processes lie at the heart of ageing . Equations of motion –describing the movement of a material system in ambient space, see Bažant, Cedolin (1991), Béda, Kozák, Verhás (1995)– lie at the heart of any physical theory. When considering motion in a coordinate system anchored in the ambient space, the coordinate system is called the Eulerian Coordinate System ( ECS ) and the description is called Eulerian. Examining motion in a coordinate system that is co-moving with the material, one speaks of a Material –or Substantial or Lagrange– Coordinate System ( MCS ) and the description is called Material –or Lagrange–. Given that the DSC s and SIC s of LSS s have been based on Continuum Field Theories ( CFT s) from the early days of safety engineering, the starting point will be kept unchanged to preserve the continuity of the theoretical framework. CFT s aim to describe the behavior of continuously distributed materials –in the chosen order of approximation– with acceptable accuracy. Material Bodies ( MB s) are viewed as being composed of elementary cells, called Representative Volume Elements ( RVE s). Within RVE s, physical quantities describing the material and its properties are thought to be characterized by sufficiently smooth distributions. A material point ( P ) is defined as the selected point of the RVE , the motion of which –or the value of the physical fields defined at this point– represents the average motion –or the characteristic values of the physical fields– of the RVE . Within the RVE , the motion field and the physical fields are expressed as P -centered local Taylor series expansions of corresponding quantities in an appropriate order. The material point ( P ) is characterized by its position ( X ) in the MCS . In this description, the MCS is the primary coordinate system, chosen because the theoretical questions are the easiest to be answered within the MCS . The ideas described here are based on the view used in the Geometric Approach to Continuum Mechanics and in the theory of Mechanics on Material Manifolds, where the material body is considered as a Material Manifold with its own Geometry –see e.g., Yavari, Marsden and Ortiz (2006), Maugin (2009)–. Due to the limited space available, further technical details are omitted. The motion of P of a MB in ambient space is described by the equation of motion as follows: ( ) ( ) ˆ ˆ ˆ , , , where d τ τ τ χ τ τ τ τ = x X (1) where X stands for the position of P in the MCS , while x in ECS ; τ marks the current time coordinate, while ˆ τ is the accumulated time since the system start-up 0 τ . The introduction of ˆ τ indicates that in the description, the motion is also observed in long-term. Hereafter, the gradient is referred to as X ∇ in MCS , and ∇ in ECS . The form of the equations of motion in (1) is the same as that in point mechanics. The form of the equation of motion in the 3.1. Fundamental considerations, basic equations 0 τ = 