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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where ρ means density, ε is linearised strain tensor, σ is the engineering stress tensor,   v means acceleration, f denotes density of external forces, W is stored mechanical energy, W is mechanical power, ∙ is inner multiplication of two vectors while : means inner multiplication of two tensors, ( ) E E d = ε F σ is elastic and ( ) p p d = ε F σ is plastic constitutive law. The approach that will be part of the advanced methodology for future SICs , uses the original – nonlinear – form of the Green-Lagrangian deformation tensor – in terms of the displacement gradient as ( ) 1 2 T T = + +  E H H H H – , and requires therefore a careful distinction between the Eulerian and the Lagrangian description. The governing equations describing the behaviour/evolution of the system within this formulation – i.e., the kinematic equation, the deformation decomposition theorem, the balances and the constitutive laws – are as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 , , kinematic equation , , , strain decomposition theorem , , , balance of momentum , , balance of moment of momentum , , : , balan X X X X E p X T d d d d W                =  + +    = + +  = = = ε X u u u u X ε X ε X ε X v X σ X f X σ X σ X X σ X ε X ce of mechanical power (2) where ρ is mass density, ε is the Green-Lagrange strain tensor – in terms of the displacement vector – , σ represents the true stress tensor, d  v means acceleration, f denotes density of external forces, W is the stored mechanical energy, W denotes mechanical power, ∙ is inner multiplication of two vectors, : means inner multiplication of two tensors , ( ) E E d = ε F σ represents the elastic, ( ) p p d = ε F σ the plastic constitutive law. Concerning the constitutive relations in (1) and (2), it is to be noted that these do not include an explicit time dependence; this means that the material model assumes an instantaneous response to loads. Having a closer look to the above two sets of equations and comparing them, it is easy to see that although the field equations are very similar in form, there is a fundamental difference between them. Relation (1) is based on a linearised theory of deformation, whereas relation (2) is a second-order theory. For small displacements and deformations, the two theories lead to nearly identical results, but for large displacements and deformations, where nonlinear components of the deformation – the geometric nonlinearities – become increasingly important, the two theories will lead to significantly different results. Moreover, since deformation in second-order theory is inherently nonlinear, any other physical quantity depending on the deformation is also inherently nonlinear. 2.2. Conceptual design considerations for the measurement and evaluation framework As explained above, model (2) is nonlinear, describing geometric nonlinearity through ( ) ( ) X X    u u and material nonlinearity through the constitutive equations. In the small deformation range the kinematic model is slightly nonlinear, and becomes strongly nonlinear in the large deformation range, where geometric nonlinearities carry important information. From physics point of view, a nonlinear theory allows a more general and deeper understanding of the system behaviour, and a proper nonlinear theory is essential to accurately represent material behaviour of structural materials in situations where strains are inevitably large, e.g., in conditions before final failure – see Bažant and Cedolin (1991), Clayton (2011), Hirschberger (2008) – . In the presence of large deformations, the non-linear behaviour of the system is governed by the geometric and material non-linearities together, in an inseparable and entangled way – or in other terms, in synergistic cooperation of the geometric and material non-linearities – . This fact has far-reaching consequences for the understanding and evaluation of measurements – even in the case of tensile tests – : the experiments should be designed carefully and instrumented with appropriate sensors, to follow with sufficient accuracy the patterns and their evolution – which characterise the nonlinear behaviour of the observed system, i.e., the test specimen – during measurements. This implies that the experiments should collect considerably more and a significantly higher quality of information on the behaviour of the system than that required by current rules of material testing standards – see e.g., ISO 6892-1 (2019.) – . ( ) E ( ) p , constitutive relations E p d d = ε F σ = ε F σ