Least-squares fitting of the linear Mohr envelope

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Abstract
Introduction Mohr's hypothesis proposes that when shear failure along a plane takes place, the normal stress σ and the shear stress τ acting on that plane have a characteristic functional relationship. This function relating τ and σ, it is proposed, depends on the material and can be represented on the στ plane by a line defining the critical values of α and τ for shear failure. In practice this critical line is constructed tangen-tially to Mohr circles representing different combinations of principal stresses applied to specimens of a particular material and is therefore referred to as the Mohr envelope. For some materials a straight Mohr envelope with the equation τ = c + μσ appears from the results of triaxial testing. Furthermore in the routine testing of some materials (e.g. soils) a straight line envelope is sometimes assumed a priori. In such tests small deviations of the Mohr circles from the envelope are attributed to errors and a best-fitting straight line is used to obtain the parameters (μ, c ) necessary to characterize the properties of the material. We describe here a procedure for calculating a best-fitting straight Mohr envelope from data consisting of the applied principal stresses (i.e. from the Mohr circles). The concept of best-fit used The criterion used for selecting the envelope of best fit is illustrated in Fig. 1. By means of a least-squares fit we represent the Mohr envelope by a straight line τ = c + μσ- subject to the condition that S

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