Damage Analysis of a Cracked Nanobeam as predicted by Scale Invariant Criterion

The rapid advances in nanotechnology, nanomaterials and nanomechanics will make manufacturing technologies and infrastructure more sustainable in terms of reduced energy usage and environmental pollution. Recent advances in the research community on this topic have stimulated research activities in science and engineering devoted to their development and their applications. A macroscopic and microscopic material damage description ahead of a crack in a single formulation that satisfies the continuum mechanics axioms with consistency has been made by Sih & Tang [1,2,3]. Different order and strength of singularity are uniquely associated with the boundary conditions, loadings and geometries of the defects under consideration. The character of the volume energy density function was found to be fundamental in scale shifting. To this end, the energy density function for the dual scale model has been determined and discussed in connection with what was emphasized. The salient features of the model can be summarized as follows: • Singularity representation is applied to simulate the effects of loading, boundary constraint and geometry at each scale level. • Equilibrium mechanics is used in each segment of scaling such that the error of approximating non-equilibrium can be controlled. • Connection can be made from segment to segment by a scale invariant criterion that corresponds to the quantity of “force”. • The one-dimensional line crack model can be extended to two-dimensions by application of the volume energy density function in conjunction with the introduction of a length or area parameter. Recently, the analysis of validity of the continuum beam models for the constitutive behaviour of carbon nanotubes and nanorods, and other nano-beams of non-carbon materials has been presented by Wong et al.[4] and Harik [5], among others. It is a common notion that small whiskers have strengths considerably greater than those observed in macroscopic crystals. The increase in strength is normally attributed to a reduction in the number of defects that lead to mechanical failure. The crystal may indeed contain more defects but if they are distributed uniformly the strength can be high. The strength depends upon the degree of system homogeneity that can be represented by a characteristic length parameter [3 ]. In this paper, the scale invariant criterion is applied to cracked nanobeams to describe material damage. It makes use of the strain energy density factor, S, which is a function of the stress intensity factors [10]. As stated above, the strain energy density theory provides a more general treatment of fracture mechanics problems by virtue of its ability in describing the multiscale feature of material damage and in dealing with mixed mode crack propagation problem. A simple method for obtaining approximate stress intensity factors is also applied [7]. It takes into account the elastic crack tip stress singularity while using the elementary beam theory. Some basic loading conditions can be studied.