Stress concentration and size effect in fracture of notched heterogeneous material

Abstract

We study theoretically and experimentally the effect of long-range correlations in the material microstructure on the stress concentration in the vicinity of the notch tip. We find that while in a fractal continuum the notch-tip displacements obey the classic asymptotic for a linear elastic continuum, the power-law decay of notch-tip stresses is controlled by the long-range density correlations. The corresponding notch-size effect on fracture strength is in good agreement with the experimental tests performed on notched sheets of different kinds of paper. In particular, we find that there is no stress concentration if the fractal dimension of the fiber network is D≤d-0.5, where d is the topological dimension of the paper sheet.

Reply to “Comment on ‘Hydrodynamics of fractal continuum flow’ and ‘Map of fluid flow in fractal porous medium into fractal continuum flow’ ”

“…such that the power-law behavior M(W ) ∝ L D is attained only when L ξ 0 , as is observed in experiments with fractal materials [9,20].…”

“…[1-7, [15][16][17][18][19][20][21]). Alternative models lead to very different solutions of the same problems for physical fractal media.…”

“…Such an approximation permits one to study physical phenomena in fractal media within a continuum framework (for reviews, see Refs. [1][2][3][4][5][6][7][11][12][13][14][15][16][17][18][19][20][21][22][23][24]). …”

From fractal media to continuum mechanics

“…This is accompanied by a split of total Cauchy and micropolar stresses into their quasi-conservative and dissipative parts 39) along with relations between the internal quasi-conservative and dissipative stresses…”

“…The material parameter γ is conventionally taken as constant, but, given the presence of a randomly microheterogeneous material structure, its random field nature is sometimes considered explicitly [37,38]. Recognizing that the random material structure also affects the elastic moduli (such as E), the computation of E e and G in (5.9) needs to be re-examined [38]; see also [39] in the context of paper mechanics. With reference to Fig.…”