BIEM analysis of a graded nano‐cracked elastic half‐plane under time‐harmonic waves

Abstract

This work addresses the scattering of time‐harmonic waves by a finite, blunt nano‐crack in a graded isotropic bulk material with a free surface. The mechanical model is based on classical elastodynamic theory and describes the elastic, isotropic, graded half‐plane with quadratically varying material parameters along the depth coordinate. This model is extended to nano‐mechanics by using non‐classical boundary conditions and localized constitutive equations at the interface between crack and matrix material following the Gurtin‐Murdoch surface elasticity theory. The computational technique employed is based on the non‐hypersingular traction boundary integral equation method (BIEM) using a closed form Green's function for a half‐plane consisting of a functionally graded material (FGM). The dependences of the diffracted and scattered waves and of the local stress concentration fields on key problem parameters such as size and depth of the embedded crack, surface elasticity effects, type and characteristics of the incident wave, and the dynamic interaction phenomenon between the crack and the free‐surface boundary are all examined through parametric studies.

Numerical Solution of Integro-Differential Equations Modelling the Dynamic Behavior of a Nano-Cracked Viscoelastic Half-Plane

“…This numerical method is based on solution of an integro-differential equation along the nano-crack interface, with kernel functions which are derived analytically in the form of a Green's function and its derivatives for the inhomogeneous half-plane, see R a n g e l o v and M a n o l i s [13]. The study is a continuation of the authors' previous results, see R a n g e l o v, D i n e v a and M a n o l i s [12], where the dynamic nano-crack problem is solved in a viscoelastic anisotropic plane, and in R a n g e l o v, D i n e v a and M a n o l i s [11], where the dynamic nano-crack problem is solved in an elastic isotropic graded halfplane.…”

“…If 0 cc  , then the blunt nano-crack has a crack root of semi-circular shape with radius c0, while if c = 0, the blunt nanocrack degenerates to a line crack with a length 2a. The blunt nano-crack's line S can be expressed in the following way, see [11]   , 0 0   are reference values for the shear modulus and the mass density, respectively, which define the corresponding homogeneous half-plane when In what follows, we define the BVP in the frequency domain in terms of the governing equations of motion and the boundary conditions.…”

“…The numerical technique developed for solving the IDE of Equation ( 9) for the nanocracked, viscoelastic, graded bulk material is based on the free surface, half-plane Green's function, see R a n g e l o v, D i n e v a and M a n o l i s [11]. The following procedural steps are now defined: (a) Discretization of the crack surface and approximation of the nodal unknowns.…”

“…, where SV  is the SV-wave length and BE l is the maximum size of the Boundary Elements (BE) used. Detailed verification studies for a nano-crack in a graded half-plane and in an viscoelastic of Zener type material under time-harmonic plane waves are presented in the authors' publications of R a n g e l o v, D i n e v a and M a n o l i s [11,12].…”

Dynamic response of elastic anisotropic solid with nanoinclusions

“…Atomistic models explicitly describe the individual atoms during their dynamic evolution (Robertson et al [4], Garg and Sinnott [5], Belytschko et al [6]), while the molecular dynamics (Bao et al [7]) takes into consideration the interactions occurring at the material microstructure. Continuum mechanics based models extending the range of classical continuum mechanics by bridging its basic theoretical principles with the most fundamental effects observed at the nanolevel are as follows, see Manolis et al [8]: the higher order and non-local elasticity models (Thai et al [9], Sladek et al [10,11]) and the surface elasticity ones based on the pioneering work of Gurtin and Murdoch [12,13] and Gurtin et al [14], see Parvanova et al [15][16][17][18][19], Rangelov et al [20][21][22], Dineva et al [23,24]. The Gurtin-Murdoch theory was motivated in part by empirical observations pointing to the presence of compressive surface stress in certain types of crystals.…”