PSI - Issue 37

Dániel Antók et al. / Procedia Structural Integrity 37 (2022) 796–803 Dániel Antók, Tamás Fekete et. al.: Evaluation Framework … / Structural Integrity Procedia 00 (2019) 000 – 000

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decades, SICs for most investigated industrial LSPSs have also been carried out following the methodologies and rules prescribed in the related design standards. As SI projects for an LSPS can be based on more knowledge of system specific properties – since the data describing the geometry of the manufactured and assembled system and the results of material tests and other relevant tests are available – and the time frame for SICs is longer than for DSCs , potentially the SICs can be performed using a more advanced – state-of-the art – methodology. This more advanced methodology for SICs can be introduced into industrial engineering projects, once a theoretical model for the methodology has been developed – based on current understanding of the state of the art necessary for the field, see e.g., Bažant and Cedolin (1991) and Maugin (2009) – , and the required material tests necessary to implement the methodology have been developed, combined with the theoretical-model-based methodology for evaluating the experiments – see Bažant and Cedolin (1991) – , and the methodology has been validated. The required material tests should be done following the standards for implementing and conducting them and must in any case be evaluated in the context of the theoretical model for the proposed methodology – which is understood as being consistent with the characteristics of the theoretical model and the material models used in it – . Given that the measurement results are intended to be used in SI problems, below is a summary describing the theoretical relationships – in a nutshell – of two different theoretical approaches to SICs that serve as a basis for the evaluation of material tests – e.g., tensile tests – , according to the chosen approach. The first approach has been widely used in standards based SICs , while the second approach will be part of the advanced methodology for future SICs . 2.1. Continuum mechanics fundamentals of the measurement and evaluation framework The strength and SI problems of LSPSs are formulated in the context of Continuum Mechanics of Solids ( CMS ). Modern continuum mechanics of solids is a very general scientific framework from which engineering models are derived using various approximations and then these models are used to solve practical engineering problems. When an engineering methodology is considered for further development, it makes sense first to review how – i.e., by which approximations – the current engineering model has been derived from the scientific theory. Hence, we now review how the standard based SI methodology of LSPSs , and the foundations of an improved methodology, can be derived from CMS . Given a solid body, its deformation in the ambient space is described by the mapping ( ) ( ) ,   = x χ X , x denoting the positions of its Representative Volume Elements (RVEs) in Euler, and X in Lagrange coordinates,  representing time.  denotes the gradient in Euler, X  in Lagrange picture. RVEs are considered point-like objects. ( ) ( ) ( )    = − u x X is the displacement vector of a point P . To remain within the classical Boltzmann approximation, ( ) ( ) , ,   =  X F X χ X is the deformation gradient around P . T = C F F is the right Cauchy-Green tensor and ( ) 1 2 = − E C I denotes the Green-Lagrangian strain tensor – see Bažant and Cedolin (1991) , Hirschberger (2008) – . If u is primary variable, displacement gradient X X =  =  H u u enters the picture that is related to F by = − H F I . With this notation, the Green-Lagrangian strain tensor is ( ) 1 2 T T = + +  E H H H H . For small displacements,  x X , the distinction between Euler and Lagrange description is unnecessary; therefore: (1) X  =  , and (2) linearised strain, i.e., ( ) 1 2 T = + E H H can be used. Standard-based methods for design of LSPSs and standard-based SIC methods are based on the linearized kinematic model and the linearized strain. In this approach, basic equations describing the behaviour of a system – i.e., the kinematic equation, the deformation decomposition theorem, the balances and the constitutive laws – are the following: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 , , kinematic equation , , , strain decomposition theorem , , , balance of momentum , , balance of moment of momentum , , : , balance of mechanical power E p T d d d W                =  +  = +  +   = = = ε x u u x ε x ε x ε x v x σ x f x σ x σ x x σ x ε x (1)

( ) p

( ) E

,

constitutive relations

= ε F σ

= ε F σ

d

d

E

p