Download PDF by John Jaeger, N. G. Cook, Robert Zimmerman: Fundamentals of Rock Mechanics, Fourth Revised Edition

By John Jaeger, N. G. Cook, Robert Zimmerman

ISBN-10: 0632057599

ISBN-13: 9780632057597

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Extra resources for Fundamentals of Rock Mechanics, Fourth Revised Edition

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These extreme values of the shear stress are denoted by τ1 , τ2 , and τ3 , respectively. As the shear stresses are zero on the planes of the principal normal stresses, the extreme values found above are local maxima. Thus far, we have found the magnitudes of the maximum shear stresses and the unit normals of the planes on which they act. To find the directions within those planes in which the shears act, consider one of these planes, and let t be the unit vector pointing in the direction of this shear traction.

Xy θ θ x ex yx (a) (b) yy 18 Chapter 2 the slanted face of the element is n = (nx , ny ). We can construct another unit vector t, perpendicular to n, which lies along this face. Being of unit length, the components of t must satisfy the condition t · t = (tx )2 + (ty )2 = 1. The orthogonality of t and n implies that t · n = tx nx + ty ny = 0, which shows that t = ±(ny , −nx ). Finally, if we require that the pair of vectors {n, t} have the same orientation relative to each other as do the pair {ex , ey }, the minus sign must be used, in which case t = (−ny , nx ).

The principle of conservation of linear momentum leads to three further equations that must be satisfied by the stresses. 5, control the rate at which the stresses vary in space. However, much useful information about the stress tensor can be derived prior to considering the implications of the equations of stress equilibrium. Of particular importance are the laws that govern the manner in which the stress components vary as the coordinate system is rotated. 5. 3 Analysis of stress in two dimensions Discussions of stress are algebraically simpler in two dimensions than in three.

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Fundamentals of Rock Mechanics, Fourth Revised Edition by John Jaeger, N. G. Cook, Robert Zimmerman

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