Abstract
Zaera et al. (Int J Eng Sci 138:65–81, 2019) recently showed that the nonlocal strain gradient theory (NSGT) is not consistent when it is applied to finite solids, since all boundary conditions associated to the corresponding problems cannot be simultaneously satisfied. Given the large number of works using the NSGT being currently published in the field of generalized continuum mechanics, it is pertinent to evince the shortcomings of the application of this theory. Some authors solved the problem omitting the constitutive boundary conditions. In the current paper we show that, in this case, the equilibrium fields are not compatible with the constitutive equation of the material. Other authors solved it omitting the non-standard boundary conditions. Here we show that, in this case, the solution does not fulfil conservation of energy. In conclusion, the inconsistency of the NSGT is corroborated, and its application must be prevented in the analysis of the mechanical behaviour of nanostructures.
Highlights
The origin of higher order continuum theories can be found in the second part of nineteenth century and in the beginning of twentieth century, the explosive growth of the application of nanostructures in several scientific and technological fields has renewed the interest on developing this kind of approaches
Other widely used approaches to address the mechanical behaviour of nanomechanical systems fall into the nonlocal continuum mechanics framework
Given the large number of works using the nonlocal strain gradient theory (NSGT) being currently published in the field of generalized continuum mechanics, it is pertinent to evince the shortcomings of the application of this theory
Summary
The origin of higher order continuum theories can be found in the second part of nineteenth century and in the beginning of twentieth century, the explosive growth of the application of nanostructures in several scientific and technological fields has renewed the interest on developing this kind of approaches. Romano et al [21] proposed the so-called stress-driven nonlocal model, in which the elastic strain at a certain point in the solid is related to the stress at all points of the domain by a convolution integral with a smoothing kernel of the Helmholtz-type This approach allows to obtain consistent solutions corresponding to the mechanical behaviour of several kinds of nanostructures [22,23,24,25,26]. We analyse the bending behaviour of a supported Bernoulli– Euler beam subjected to an uniformly distributed static load, which was considered in different works [39, 44, 46] Through this straightforward example we show that: (i) if constitutive boundary conditions are omitted, the equilibrium fields are incompatible with the constitutive equations of the material, as it was pointed out by Barretta and de Sciarra [40]; (ii) if non-standard boundary conditions are omitted, the.
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