Complete General Solutions for Equilibrium Equations of Isotropic Strain Gradient Elasticity

Higher order asymptotic crack-tip fields in simplified strain gradient elasticity

“…Thus, general solution for the equilibrium equations of simplified SGE ( 12) can be represented through the classical harmonic vector Φ Φ Φ and harmonic scalar φ and three additional gradient scalar stress functions ψ, χ, χ. Formal derivation and proof for the completeness theorem of representation ( 13)- (15) for the isotropic Mindlin-Toupin strain gradient elasticity have been presented recently in [41]. Application of general solution ( 13)-( 15) to crack problems has been suggested in our recent work [42], though in the present study we derive its final relations in more useful separable form for the higher order cracktip fields.…”

“…Papkovich-Neuber (PN) general solution for equilibrium equations (12) can be presented in the following form [11,41]: (13) in which u u u c and u u u g can be treated as the classical and the gradient parts of the displacement field, respectively, κ = 1/(4(1 − ν)) is standard coefficient that persists in classical PN solution [55], r r r is the position vector, and the stress functions Φ Φ Φ(r r r), Ψ Ψ Ψ(r r r), φ(r r r), ψ(r r r) have to satisfy:…”

“…Let us show, how do the classical singular terms can be formally avoided from SGE asymptotic solution. We consider the symmetric loading and put n = 1 in ( 39), (41) to obtain:…”

“…Let us show, how the higher order terms can be derived within SGE by using general solution (38), (39), (41). In the above considerations, we used the terms with n = 1 (classical vanished terms) and n = 3 (leading order terms).…”

“…In the present study, we consider the problem of derivation of an explicit solution for the higher order asymptotic in-plane crack-tip fields within the simplified SGE with single additional length scale parameter [37][38][39][40]. This solution will be obtained by using the variant of Papkovich-Neuber (PN) general solution for SGE equilibrium equations suggested in our recent works [11,41,42]. Previously, asymptotic solutions for the leading order terms within SGE with simplified and general constitutive equations have been derived in Refs.…”