![]() derived the explicit bending deflections of cantilever nanobeam under concentrated force and moment at the free end using SGM, and calculated energy release rate and stress intensity factor of mode I crack for the edge-cracked nanobeam and centrally-cracked nanobeam. Therefore, the high-order continuum mechanical models, including strain gradient model (SGM), , couple stress model (CSM), , and nonlocal model, , are utilized to study the size-dependent mechanical behaviors of nanostructures, ,, ,, ,, ,,. Through the advance of the nanotechnology, it is still extreme difficulty to study the size-dependent fracture behavior of nanobeams, one of the simplest nanostructures, using molecular dynamics simulations or experimental tests. Therefore, it would be of great interest for both engineering and science communities to understand of the nanoscale fracture mechanisms. Experimental tests, ,, and atomistic simulations, ,, have shown that the fracture behavior of nanobeams is size dependent, and the superior fracture performance of nanobeams is obtained compare to macrobeams. Nowadays, nanobeams are widely utilized in nano-electromechanical system (NEMS). The size-dependent fracture behavior can explain the superior fracture performance of nanomaterial to some extent. The influence of nonlocal parameters on the size-dependent fracture behaviors including external work and energy release rate as well as the accuracy of asymptotic solutions for energy release rate is investigated numerically. In addition, the asymptotic solutions for energy release rate are deduced for small nonlocal length parameter. Accordingly, one can calculate the work done by external loads and energy release rate for both mode I and II crack problems. The bending deflections are deduced and expressed in explicit form for different boundary and loading conditions. SD-TPNIM with symmetrical or anti-symmetrical conditions is applied to formulate the size-dependent fracture behavior of simply-supported and clamped–clamped centrally-cracked nanobeams subjected to uniformly distributed load (UDL) or middle point force (MPF) along opposite (Mode I crack) or same (Mode II crack) directions. Compare with classic SD-TPNIM, two extra constitutive continuity conditions are introduced, and a few integral items are introduced to constitutive discontinuity conditions and constitutive boundary conditions at symmetrical point. A novel mathematical formulation is presented to deal with stress-driven two-phase local/nonlocal integral model (SD-TPNIM) with discontinuity and symmetrical conditions, which is transformed equivalently into the differential form together with four constitutive constraints.
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