2. Coulomb’s law of failure

Hydraulic fracturing results from a pulling apart or tensile failure of the rock mass, but natural earthquakes are caused by sliding or shear failure along a plane. Many data on shear failure under compressive stress are available from the results of triaxail tests. (In the Earth below 100- to 200-m depth all stresses are compressive). In triaxial tests a piston apparatus is used to place a cylindrical volume of rock under a directed compressive stress that defines σ₁, while the value of σ₃ is controlled by the fluid pressure in a surrounding vessel. For any given value of σ₃, the value of σ₁ can be increased until the sample fails along a plane located at some angle θ to the axis of compression (Figure 5.2a). For many geologic media θ ~ 30°.

For a uniform confining pressure σ₃ it is possible to plot the stress conditions at failure on a two-dimensional Mohr diagram, a graphic representation of the state introduced by the German engineer Otto Mohr in 1882. The Mohr circle defines the normal (σ_n) and shear (τ) stresses on the failure plane itself, provided that σ₁, σ₃, and θ are known, as they are in the case of failure in triaxial tests (Figure 10.2b). A Mohr circle with radius (σ₁-σ₃)/2 is centered on the abscissa at (σ₁₊σ₃)/2. A line is drawn from the center of the circle at an angle of 2θ relative to the abscissa. The intercept between this radial line and the circle defines σ_n and τ on the failure plane, that is,

(5.5)

and

(5.6)

Under compression the values σ₁ and σ₃ are both taken as positive, as shown on Figure 10.2b. Under pure tension, σ₁ is positive and σ₃ is zero. Under pure shear, σ₁ is positive and σ₃ is negative.

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Thus, the position of the Mohr circle along the abscissa indicates the type of stress state. The size of the circle indicates the magnitude of the difference between σ₁ and σ₃ and is a measure of the differential stress.

For numerous experiments, conditions at failure are well described by Coulomb’s law of failure,

, (5.7)

where φ is called the angle of internal friction and is often about 30°. Thus tan φ, which is also known as the coefficient of friction, typically has a value of about 0.6. The French military engineer and physicist Charles Augustine de Coulomb predicted this relationship in the 1770s. At that time it was already known that

, (5.8)

described the behavior of noncohesive materials such as sand; Coulomb’s insight was to add the constant τ₀ to represent the intact shear resistance of a cohesive material. The Coulomb failure line described by Eq. (5.7) bounds a region of stability on Figure 5.2b Mohr circles that touch the failure line describe conditions required to induce failure.

Although Coulomb’s law of failure describes many experimental results well, it remains a fundamentally empirical relation, because it is not yet possible to predict macroscopic failure behavior from molecular-scale behavior or from first principles (e.g., Scholz, 1990; Marder and Fineberg, 1996).

An apparent discrepancy between the failure behavior of drained and undrained fluid-saturated samples troubled early students of rock mechanics. In “drained tests” on saturated material, applied stresses are increased slowly and water is allowed to escape from the sample, and the results were similar in most respects to those obtained for dry samples. “Undrained tests”, in which the contained water is retained by an impermeable jacket, gave apparent φ values of ~0° (i.e., the failure criterion of Eq. (5.7) was defined by a horizontal line). By the 1950s it was recognized that the apparent discrepancy between drained and undrained behavior disappeared if, in the case of saturated samples, effective stresses were plotted on the abscissa of the Mohr diagram (i.e., σ₁ – P_f , σ₃ – P_f).

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To summarize, then, in triaxial tests any sample can eventually be made to fail by increasing σ₁ with σ₃ held constant (Figure 10.3a) and/or by decreasing σ₃ with σ₁ constant (Figure 10.3b). Fluid-saturated samples can also be brought to failure by increasing the internal fluid pressure P_f . Varying σ₁ or σ₃ can induce failure by increasing the differential stress on the sample, an effect expressed on the Mohr diagram as an increase in the size of the Mohr circle. Increasing fluid pressure does not affect the shear stress but induces failure by decreasing the effective strength of the sample. This is expressed by shifting the circle toward the origin without changing its size (Figure 5.3c).

Figure 5.3 Mohr diagrams showing failure conditions induced by (a) increasing the greatest principal stress σ₁, (b) decreasing the least principal stress σ₃, and (c) increasing the internal pore-fluid pressure P_f. In each case the σ₁ and σ₃ values plotted on the abscissa are actually the effective stresses, that is stress σ₁ - P_f and σ₃ - P_f.

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Glossary ” Relationship between hydrogeology and seismic activity”.

effective stress	действительное напряжение, эффективное напряжение
to credit	приписывать (кому-л. совершение какого-л. действия)
Coulomb's law	закон Кулона
downward force	сила, направленная вниз
tensile strength	предел прочности на разрыв
parting	поверхность стыка, отделение; разделение; разветвление; междупластье; тонкий прослой породы, пропласток
overburden weight	давление покрывающих пластов
tensile failure	разрушение при растяжении; разрыв
mutually orthogonal	взаимно ортогональный
orthogonal	ортогональный, прямоугольный
principal stresses	главные напряжения
thrust-fault	надвиг
deviator stress	девиаторное напряжение
arbitrary	произвольный, случайный
aligned	выровненный, ориентированный
off-diagonal	недиагональный
vigorous	сильный, энергичный; решительный
indistinguishable	неотличимый ( from)

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shear failure	разрушение при сдвиге, разрушение при срезе
triaxial	трёхосный
piston	поршень
vessel	1) сосуд (для жидкости); 2) корабль, судно; летательный аппарат
abscissa	абсцисса; ось абсцисс
relative	относительный, сравнительный, взаимный (о расположении)
intercept	пересечение
insight	проницательность; способность проникновения в суть ( into)
intact	нетронутый; незатронутый, неповрежденный, невредимый, целый Syn: entire
region of stability	область стабильности
discrepancy	разница; различие, несходство, отличие (between; in) Syn: difference
disclosed	раскрытый
annual	ежегодный; годичный, годовой