Energy Criterion for Fracture of Rocks and Rock-like Materials on the Descending Branch of the Load–Displacement Curve

Abstract

This article deals with the problem of predicting the brittle fracture of rocks and similar materials, which can also include frozen sandy soils. Such materials, due to the diversity of their conditions of origin, are characterized by natural heterogeneity at the micro-, meso-, and macro-levels, which makes it difficult to develop sufficiently universal criteria for their strength. Despite a number of known models and criteria of strength and fracture, the search for such criteria remains an urgent problem. In this paper, using the energy approach to the mathematical modeling of mechanical systems, the fracture criterion is justified, which differs from the known criteria that do not require integration to calculate the strain energy We and dissipation energy Wd. The well-known relation for the input energy W=We+Wd is used. The object of the study was the ratio of dW=dWe+dWd. The main research question concerned what the ratio of dWe and dWd would be at the point of brittle failure. The search for an answer to the question led to the justification of a differential energy criterion for the failure of brittle materials on the descending branch of the full stress–strain curve. It was found that the point of predicted fracture is determined by the equality σ=0.5 εEtangential (if there is an inflection point on the ascending branch) or σ=0.5 εEsecant_secant. The main result of the work was ascertaining the differential strength and fracture criteria of brittle materials in the form of inequalities and equations, which were oriented for application in engineering calculations. Examples of application of the developed criteria are given; their consistency with the experimental data known from the literature confirmed.

Differential Energy Criterion of Brittle Fracture as a Criterion for Wood’s Transition to the Plastic Deformation Stage

“…Figures 8 and 9 explain the application of the differential brittle fracture energy criterion [45] to the analysis of the above-mentioned wooden structure (Figure 6) [38]. Figure 9 shows that the transition to the plastic stage is realized at a load of 100 kN, which agrees with the experimental data from [38].…”

“…Since we consider timber as a structure [4], it is permissible to compare the dependence according to Figure 4 with the results of tests of wooden structures known from the literature [38], according to which 𝑆 . 45.1 ; 𝑆 . .…”

“…The goal of this part of the work can be formulated as a search and substantiation of the criterion for the transition of wood from the elastic stage of deformation to the plastic behavior, i.e., from stages 3-4 to stages 4-5, according to Figure 2. Taking into account the external similarity of the ascending and (partially) descending branches of the complete load-displacement curve in tests of wood and brittle materials, consider the application of the recently proposed differential energy criterion of brittle fracture [45]. According to this criterion, in the context of this work, the point of brittle failure is located at the intersection of the post-peak branch of curve (1) and line (13):…”

“…Figure 7 explains the application of the differential brittle fracture energy criterion [45] and the tangential stiffness (8) to the analysis of the aforementioned wooden structure (Figure 6) [38].…”

Study on Rock Failure Criterion Based on Elastic Strain Energy Density

“…Wang et al [11] conducted conventional triaxial compression tests on marbles and sandstones with different bedding plane dip angles under different confining pressures, obtained the influences of confining pressure on critical strain energy release rate and energy transformation of anisotropic intact rocks, and established a new critical strain energy release rate failure criterion. Gennady et al [12] use the energy approach in the mathematical modeling of mechanical systems; the fracture criterion does not require integration to calculate the strain energy and dissipation energy. The aforementioned studies primarily analyzed the energy types and evolution laws during the deformation and failure of rocks under different conditions of stress; However, most of the existing research establishes rock failure criteria from the mechanical point of view, and few studies establish rock failure criteria from the relationship between peak elastic strain energy density and confining pressure.…”