Phenomenological criteria for the strength of brittle materials

Abstract

The accuracy of calculations for bearing capacity of objects made of brittle materials depends to a considerable degree on the strength criteria used. Most widespread among them in engineering practice are statistical, mechanical, and phenomenological criteria only connecting strength with stress state. The achievements and disadvantages of this approach are well known. It is notes that recently in crack mechanics these criteria are being used more frequently for evaluating local strength [1]. As has been established, strength criteria for brittle materials in space u~, u2, and us describe surfaces open to the direction of compressive stresses symmetrical relative to the spatial diagonal u~ = o2 = us and intersecting this diagonal at a point of uniform triaxial tension. Recent experience has confirmed [2] that limiting surfaces, at least in the region of the starting coordinate, are shapes half way between a cone and a paraboloid, and they are deformed from three sides; the greatest deformation is observed close to the tip of the shape so that the deviator sections in this zone are close to an equilateral triangle, and consequently, limiting surfaces may be described by the first theory for strength. With an increase in the hydrostatic components of the stress tensor, buckling of deviator sections increases until appearance of an obvious tendency to grow into a circle or an equal-sided hexagon, and meridional sections are smoothed out. These mechanisms are the most distinct of those observed in testing rocks [3][4][5] and therefore they were the first to be reflected in criteria orientated towards these materials, particularly the Estrin criteria [6].Many criteria for brittle materials are modifications of the Moore criterion (which is explained by its simplicity and clear physical meaning) which describe in one way or another individual properties of the deformed surface. Absence of accounting for o~ in the Moore hypothesis reflected in insufficient curvature for deviator sections is overcome by substituting normal stress o n in the area of maximum shear by octahedral stress Ooc, and smoothing out of meridlonal curves has been provided byprescribing different forms (fractionallinear, exponential, power, etc.) of envelope curves [4,[7][8][9]. In spite of the fact that only two material constants are used with these criteria, i.e. compressive strength o c and tensile strength of, or cohesive modulus and internal friction angle, they are in fact multiconstant criteria since in order to determine these or any other parameters for the deformed limiting surface additional test data are necessary.Another method of creating criteria interpreting deformed surfaces is introduction into energy criteria based on the Nadai hypothesis o i = f(Uoc ) (the criteria of Botkln, Mirolyubov, Belandin, etc.) of a third invariant of the stress deviator [i0] or the LodG-Nadai parameter ~ [2]. The main drawback of these criteria is the extreme cumbersomeness and impossibility solution without a computer.Combined criteria are proposed co...