not uniform. Stress distribution in the contact between an elastic half-space and sharp-

edged rigid or elastic counter-bodies of a different shape (e.g. a cylinder, frustum etc.)

fracture in the regions with heterogeneous stress/strain fields, non-local fracture criteria

have been developed. Perhaps one of the most popular nonlocal fracture criteria is

Novozhilov fracture criterion [9] which reads

, (1)

where σc is the strength of a homogeneous material under uniaxial tension without stress

concentrators, σy(x) is the maximum normal stress, x axis corresponds to a “most probable”

direction of crack propagation, d1 is a material constant having a dimensionality of length.

Similar forms of nonlocal criterion (1) were used in [10] for composites and in [11] for

notched samples. Another way of taking into account stress concentrators is to include

both absolute value of stress (either maximal or average value) and its gradient in a crack

processing zone:

concentration zone [12]. In the framework of this approach the size of stress concentration zone

depends on a local stress gradient: Le = Le(grad(σ)).

with stress concentrators, it is necessary to adequately describe crack nucleation conditions

since they significantly depend on stress distribution approximation accuracy that is

determined, in turn, by mesh size. Correspondingly, variation of the mesh size may have

influence on the values of mechanical characteristics of the simulated samples, e.g. their

compressive/tensile strength, etc.

refinement, is not applicable to problems where “quasi-infinite” local values of stresses

the boundary element method (BEM) for a normal adhesive contact problem of linearly

elastic and power-law graded elastic materials [15-17]. The main idea of the suggested

approach is a modification of local ultimate stress value instead of stress field correction.

The given criterion is local and it allows adequate descriptions of a normal contact

between an elastic half-space and a rigid indenter in the presence of adhesive forces.

continuum-based approaches like FEM, the size-dependent fracture criteria in DEM still

remain much less studied [18]. In the paper we study the relation between tensile and

shear contact strength and a discrete element size. Hereinafter the term “contact strength”

means a maximal value of a force divided to the contact square that is required to detach

punch while the lower layer of the half-space is fixed in vertical direction. Horizontal

motions of elements of the top boundary of the punch and the bottom boundary of the half-

space are allowed. The materials of the punch and the half-space are considered linearly

elastic. We used the Movable Cellular Automaton method (MCA) [19-21] which is a DEM

family representative with multi-particle distinct element formulation of the equations of

elasticity and motion.

the punch Epunch was varied in a certain range in order to perform a parametric study of the

Evidently, the boundary conditions produce a contribution to the values of sample strength

obtained during simulations. In order to diminish influence of the boundary conditions we

studied the convergence of the strength value when thickness and height of a punch and a

counter-body simulating a half-space increase. The values of thickness H and width W of

the counter-body greater than five widths of the punch 2a0 guarantee convergence of the

value of the tensile strength and, consequently, the absence of the boundaries’ influence.

Punch height h>3a0 guarantees tensile strength being independent of h. The following

values of the sizes of the punch and the “half-space” were used in the calculations

described below: a0=0.01 m, h=0.03 m, H=0.125 m, W=0.1 m. The closed package of

discrete elements was used. It should be noted that the use of a close-packed ensemble of

circular discrete elements leads to the appearance of an "artificial roughness" on the boundaries

occurs when a given fracture or linkage criterion is satisfied.

through their common plane faces. The evolution of an ensemble of discrete elements is

defined by a numerical solution of the system of Newton-Euler equations of motion:

the following form:

written for a pair of interacting discrete elements.

simulation of detachment of a punch from a half-space. Namely, tensile strength σt of a

contact between a punch and half-space becomes lower with decrease of the diameter of a

discrete element in accordance with the power law:

, (6)

element size for different relations of elastic moduli of the punch and the half-space

to zero. As one can see from Fig. 2, there is no convergence (at least, asymptotic) of the

values of tensile strength on the lower limit of element size. This means the formulation of

the numerical model is physically inadequate and must be improved in a way guarantying

convergence of the results of simulation with decreasing of an element size. It is necessary

to note the fact that the mentioned effect of element size dependence of tensile contact

strength holds for different types of fracture criteria (von Mises, Drucker-Prager, detachment

at a given value of normal tensile stress etc). Also this effect exists when a quadratic

top layer of the punch, while it was fixed in vertical direction. It was found that shear

strength of the contact between the punch and the half-space depends on the element size

in the same manner as the tensile strength discussed above (see Fig. 3).

, (7)

logarithmical dependence of shear strength on the element size. Based on the analysis of

the dependencies shown in Figs. 2 and 3 we obtained the following estimate of the exponents

from Eqs. (6) and (7) for e = 1:

. (8)

that the highest discrepancy between the values of exponents s and q takes place for e = 10

This demonstrates universality of the obtained dependencies of strength on element diameter

distribution in the contact area with high values of stresses near boundaries. The peaks of

stresses take place for all components of stress tensor and for equivalent (von Mises)

stress which determines crack formation due to the fact that the von Mises fracture

criterion was used in the performed calculations. At that, the smaller an element size the

higher the peaks and the earlier fracture occurrence (see Fig. 4).

where F – total force, applied on punch, M(λ, a) – a non-dimensional weight function, x –

spatial coordinate along the contact patch measured from the punch center, λ – the

exponent (see also, [23]). There is an analytic solution for λ, obtained by Rao [3] for a

punch with an arbitrary angle at the corner θ:

. (10)

plane strain conditions [3].

contact area, we obtained the estimation λ ≈ 0.39 (see Fig. 5). This value agrees well with

the values of exponents s and q, which enter Eqs. (6) and (7) correspondingly. The given

fact demonstrates that the character of the dependence of contact strength on element size

is determined by stress distribution in a contact area.

stress in the von Mises failure criterion:

, (12)

requires additional research. Application of Eq. (12) in fracture criterion (5) allows obtaining

almost the same values of tensile strength of samples with different element size (see Figs.

6a and 6b). One can see that the loading diagrams of samples under tension are almost

identical (Fig. 6b). A distinction between them is conditioned by a small difference of

stiffness of samples with different element size. The deviation of tensile stress from its

sample mean value does not exceed two percent (Fig. 6b).

of microscopic failures nucleation, motion and healing [22]. This is one of the facts

underlining similarity and interconnectedness of plasticity and fracture phenomena. From

this point of view it is obvious to make an assumption about an existence of mesh-dependent