PSI - Issue 42

T. Fekete et al. / Procedia Structural Integrity 42 (2022) 1684–1691 T. Fekete et al.: Extending reliability of FEM simulations… / Structural Integrity Procedia 00 (2019) 000–000

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relevance of calculations to determine the expected lifetime of engineering structures under design or in operation is increasing and will become even more important with the advent of the DT technology. Simulations that investigate the expected behavior of an equipment under design are called Design Safety Calculations ( DSC s), while simulations evaluating the behavior of operating systems are called Structural Integrity Calculations ( SIC s) –see Fekete (2019 and 2022)–. The theoretical framework used in these calculations still bears the imprint of continuum mechanics from about a century ago. In the meantime, during the last 100 years, Information Technology ( IT ) has developed, and CMS and numerical mathematics have made significant progress. These achievements –among others– have made possible ongoing developments, which are expected to result in SIC s based on a new methodology, rooted in modern CMS . It is expected that the theoretical framework proposed for the methodology to be developed will yield calculations with higher predictive power than the approach commonly used in present day calculations. The improvement expected in the accuracy and reliability of the calculations can only be achieved by considering that theory and measurements form an entangled system. This implies that the same theoretical apparatus should be used in the evaluation of the measurements required as for the subsequent SIC s –see Bažant and Cedolin (1991)–. Any physical theory has at its roots in the equations of motions –see Musilová and Hronek (2016)–, which describe the motion of a material system in its ambient space –see Bažant and Cedolin (1991), Béda, Kozák and Verhás (1995), Hirschberger (2008) and Papenfuß (2020)–. The field equations supplement the equations of motions by representing interactions within the system and their effects –see Musilová and Hronek (2016)–. The behavior of the system is described by a coupled system of equations of motions and the field equations. When the equations of motions are considered in a coordinate system fixed in the external space, the description is called Eulerian, and the coordinate system is called Eulerian coordinate system. If the motion is considered in a system that moves with the material system, the description is called Lagrange, and the coordinate system is called Lagrange/Substantial. Nowadays, primarily in the Geometric Approach to Continuum Mechanics, the material body is considered as a manifold with its own Geometry –see e.g., Yavari, Marsden and Ortiz (2006)–. Therefore, a coordinate system is introduced over the body, also called the Material/Reference Coordinate System. The material points ( P ) are characterized by their position ( X ) in the material coordinate system, which is the primary coordinate system, meaning that the material/Lagrange coordinates play a privileged/primary role. The motion of a point P of a body in external space, i.e., in the Eulerian coordinate system, is described by the equation of motion as follows: ( ) ( ) , τ χ τ = x X (1) where X denotes position of P in the material –i.e., in Lagrange– coordinate system, while x in Euler coordinates; τ represents time. This version of the motion equations is like the form used in point mechanics. A step forward is possible if it is kept in mind that the aim of continuum mechanics is to describe the collective mechanical behavior of materials phenomenologically, using a field theory based on the continuum hypothesis. The simplest formulation of the kinematical field is obtained by examining how the equations of motion change when one considers the motion of a point P ' in the small neighborhood of –i.e., in the Representative Volume Element ( RVE ) around– P . For , d ′ ′ → → = + P P X X X X , the motion of P ' –assuming the simplest locally linear form of the motion function– takes the form in Euler system of as follows: ( ) ( ) ( ) ( ) , , X d O d d τ χ τ χ τ ′ = +∇ + ⊗ x X X X X X (2) ( , ) X χ τ ∇ X is the local material deformation gradient map at X , which is locally linear, and called F from here on; the second and higher order terms are concealed in ( ) O d d ⊗ X X –see e.g., Bažant and Cedolin (1991), Yavari, Marsden and Ortiz (2006) and Hirschberger (2008)–. In the sequel, ∇ denotes the gradient in Euler, X ∇ in Lagrange coordinates. The displacement vector at P –connecting positions of P corresponding to Euler and Lagrange coordinates– is defined by: ( ) ( ) ( ) τ τ τ = − u x X (3) where x ’ denotes the position of P ' in the ambient space,