Issue 39

M. Romano et alii, Frattura ed Integrità Strutturale, 39 (2016) 226-247; DOI: 10.3221/IGF-ESIS.39.22 234         A L L A x dx A L A L L A E A L 2 2 2 2 2 2 0 2 2 2 2 2 4 1 sin 2 1 2 2 4 , 2 2 1 2                                                (12) The previously stated presumptions and boundary conditions have to be fulfilled in order to derive a kinematic due to geometrical constraints only. Therefore the arc length has to remain constant under the application of selected relative displacements u rel . For selected degrees of deformations the governing equation is       rel rel f A s u s u 1 0 0     (13) that is solved numerically in order to determine the resulting root A 1 . For each pre-selected degree of deformation u rel a respective shift in the amplitude Δ A can be calculated according to Eq. (13) by     rel rel A A u A u A A 1 0 0       (14) that is the shift in amplitude as a function of the applied relative displacements u rel . For further investigations it is reasonable to introduce the relative shift of the amplitude rel A A A A w A A A 1 0 1 0 0 0 1       (15) in order to receive another relative dimension, corresponding to the relative dimension u rel (7). The relative shift of the amplitude w rel (15) is plotted against the relative displacements u rel (7), and w rel - u rel -diagrams result. The correlation reaches its limit at a degree of deformation   rel,max s A l u l 0 0 0   (16) For this value the algorithm to numerically determine the shift in amplitude reaches a singularity. In this case the singularity represents the state when the former sine gets completely flattened by positive deformation. This leads to an absolute decay of the amplitude A . In contrast shortening as negative deformation shows a less gradient of the shift of the amplitude A towards higher values. In case of applied negative deformations the model under the previously stated presumptions is able to theoretically describe the increase of the amplitude A up to a maximum value in the degree of deformation rel u 1   . This state means a complete compression. For this value the algorithm to numerically determine the shift in amplitude exhibits its second singularity. N UMERICAL INVESTIGATIONS BY F INITE -E LEMENT -A NALYSES n order to verify the analytical model numerical investigations with the Finite-Element-Analyses (FE-Analyses) are carried out. As linear-elastic FE-calculations are carried out the elastic parts, formerly neglected in the analytical model, are considered. These are namely elasticity and transversal deformation due to Poisson effects of the model under selected longitudinal deformations as strains in terms of relative displacements. In order to identify the originally acting mesomechanic kinematic correlations due to geometric constraints, non-linearities, failure mechanisms and friction effects are neglected. The modeling, setting and processing described in the following is based on the investigations of Ottawa et al. 2012 [23]. Therein different element formulations, number of elements (degree of discretization) and I