Size-dependent fracture analysis of Centrally-Cracked nanobeam using Stress-Driven Two-Phase Local/Nonlocal integral model with discontinuity and symmetrical conditions

Abstract

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. 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. 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. The bending deflections are deduced and expressed in explicit form for different boundary and loading conditions. Accordingly, one can calculate the work done by external loads and energy release rate for both mode I and II crack problems. In addition, the asymptotic solutions for energy release rate are deduced for small nonlocal length parameter. 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. The size-dependent fracture behavior can explain the superior fracture performance of nanomaterial to some extent.