Nonlocal strain gradient finite element analysis of nanobeams using two-variable trigonometric shear deformation theory

In the present paper, a new trigonometric two-variable shear deformation beam nonlocal strain gradient theory is developed and applied to investigate the combined effects of nonlocal stress and strain gradient on the bending, buckling and free vibration analysis of nanobeams. The model introduces a nonlocal stress field parameter and a length scale parameter to capture the size effect. The governing equations derived are solved employing finite element method using a 3-nodes beam element, developed for this purpose. The predictive capability of the proposed model is shown through illustrative examples for bending, buckling and free vibration of nanobeams. Comparisons with other higher-order shear deformation beam theory are also performed to validate its numerical implementation and assess its accuracy within the nonlocal context.

     

Cyclic Softening Modeling with the Distribution of Non Linear Relaxation (Dnlr) Approach

Being of particular interest in this work, a complicated phenomenon related to cyclic softening of metallic polycrystals is modeled. As in the Waspaloy, this phenomenon can take place when a non-proportional tension-torsion cyclic loading of 90° out-of-phase is followed, after cyclic steady state, by a uniaxial one (tension-compression) with the same maximum equivalent plastic strain. By using the DNLR (Distribution of Non Linear Relaxation) model recently proposed by the authors describing the cyclic plasticity of metals, a new extension is here developed. It is recognized that such an extension can satisfactorily reproduce this softening phenomenon. It is noteworthy that this model describes such a phenomenon with a minimum number of material parameters in comparison with other phenomenological models. After calibration of the model parameters for the Waspaloy, the constitutive equations of the model are then implemented in a commercial finite element code to simulate the model response vis-à-vis a given structure made from the Waspaloy. Finally, it is found that the new extension as well as the finite element predictions give a fairly well accordance with the available experimental results.