ABSTRACT
A mathematical analogy is drawn between the terminal stages of material failure and the precursory behavior of stressed rock approaching collapse. Deterioration of materials approaching failure may be described by the simple law https://www.w3.org/1998/Math/MathML"> Ω ˙ − α Ω ¨ − A = O https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078944/0b7fd3f0-44e6-4199-ab05-800138e3be6e/content/inq_chapter3_7_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where A and α are empirical constants, and Ω is an observable quantity such as strain. The properties of this differential equation are developed and are shown to describe material behavior during the process of creep. A large amount of data from many sources conform to the proposed equations, for such materials as metals and alloys, ice, plastic, concrete, soil, as well as rock. Independently-discovered relationships emerge as special cases, including forms of the Robotnov-Kachanov, Monkman-Grant, Dobes-Milicka, Sandström-Kondyr, Saito, and exponential deformation equations. At any time t = t* where Ω = Ω*, time to failure t f is given by a simp1e expression such as the upper bound https://www.w3.org/1998/Math/MathML"> t f − t * = Ω ˙ * 1 -α [ A(α-1) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078944/0b7fd3f0-44e6-4199-ab05-800138e3be6e/content/equ_chapter3_7_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Time to failure can also be predicted by a graphical procedure using a reciprocal-rate curve; failure occurs when the extrapolated reciprocal-rate curve nearly intersects the time axis. Under constant loading, such curves decrease continuously with time; linearly for α = 2, upwardly convex for α > 2, and concave for α < 2. However, the graphical procedure may be applied generally, even for conditions of variable loading. Predictions are easily updated with new observations. Potential applications in rock mechanics include slope stability, ground control, and roof fall hazard mitigation. Case histories demonstrate practical applications.
