TWISS & MOORES, Chapter 8, Stress
Introduction: force, traction, and stress, and their relations
General physical terms include: force, a vector, where
F = ma;
traction, which represents the intensity of a force acting on an area, where
and surface stress, which has the same units as traction and can be thought of as the reaction of a given surface (size, orientation) to a pair of equal and opposite tractions. The totality of surface stresses, s , acting on a point (infinitesimal cube) is described by the stress tensor. A given stress may be separated into its components, tnacting normal to surfaceand tsacting parallel to surface (see Figure A below). The convention is that compressive stresses are considered positive and tensile stresses negative. Shear stresses ("couples") may be counterclockwise (considered positive) or clockwise (negative). This is the Mohr circle sign convention.
Neither traction nor surface stress are vectors because they depend fundamentally on area and thus on the orientation of the surface on which they act (a force, on the other hand, is independent of surface orientation). The stress ellipse is commonly used to represent the state of stress at a point, with axes proportional to the magnitude of normal stress; this is not always a useful depiction, however, because if both positive and negative stresses exist the ellipse is undefined.
The principle axes are those directions along which the shear stress components are zero and normal stresses are thus at a maximum; these axes are mutually perpendicular, and the normal stresses parallel to them are principle stresses (s 1, s 2, s 3), where by convention s 1 ³ s 2 ³ s 3. The principle planes are defined perpendicular to their respective axes. The following is a discussion of only two-dimensional stress, with principle axes x1 and x3.
The stress at a point may be completely defined in two ways:
For the case of hydrostatic (or lithostatic) pressure, only one surface stress is necessary to describe the stress state.
To fulfill the condition of steady-state equilibrium, the forces as well as the moments acting on the infinitesimal square (cube in 3-D) must sum to zero; that is, the forces and moments must consist of equal and opposite pairs. This fact may be used to show that there are only three independent stress components in a 2-D system: txx, txz = tzx, and tzz.
In 3-D, the stress state s at a point is completely described (again) by either the principle stresses and their orientations, or the surface stresses, their components, and orientations; there are nine stress components in 3-D, of which six are independent.
The stress tensor
The stress tensor s is a mathematical quantity that allows one to calculate the state of stress on a plane of any orientation through a given point. There is a different sign convention for the stress tensor, and the signs of shear stresses in particular may be different. The tensor is written as a matrix. In this, the principle diagonal notes the normal stresses, while shear stress components occur in off-diagonal positions. To maintain tensor symmetry, we accept by convention that t12 = t21, t13 = t31, and t23 = t32. (This set of relationships is opposite that of the Mohr circle conventions, and the different notation is intended to deflect confusion.) Also, in the principle coordinate system, the off-diagonal positions are equal to zero (no shear stress), and the principle diagonal components are the principle stresses (maximum normal stress).
Mohr diagram for 2-D stress
(Again, this discussion will be restricted to 2-D stress unless otherwise noted.) There is a geometric relationship between shear and normal stress components that is often difficult to visualize. Toward this end, the Mohr diagram is useful (see Figure B). Its horizontal axis is the normal stress tn, and the vertical axis shear stress ts. For a given stress, the stress states on surfaces of all orientations through the point plot (in Mohr space) as a circle, known as the Mohr circle.
The Mohr diagram has special properties. Its axes are values of stress, not position in physical space. It completely represents the state of stress at a point. Maximum and minimum normal stresses (i.e., the principle stresses s 1 and s 3) are located at the intersection of the Mohr circle with the horizontal axis; there, shear stress equals zero.
The orientation of the plane in question (angle q ) is measured between the normal (n) to the plane and the maximum principle stress; on the Mohr diagram, one plots 2q , not q . The stress components separated by 180° on the diagram are those acting on perpendicular planes in physical space (a = 90° , wherea is the angle between the planes in physical space). Stresses acting on planes oriented at q = 45° (2q = 90° ) to the maximum compressive stress are known as conjugate planes of maximum shear stress, because on them normal stress equals the mean normal stress and shear stress is at a maximum.
The magnitude of stress at a point can be characterized by two scalar invariants, the mean normal stress and the radius r, where:
mean tn = (s 1 + s 3) / 2, and
r = ç ts (max)ç = (s 1 s 3) / 2.
Normal and shear stress components may be calculated for any plane in physical space (assuming the above notations) using the Mohr circle, or the equations:
tn = [(s 1 + s 3) / 2] + [(s 1 s 3) / 2] cos 2q , and
ts = [(s 1 s 3) / 2] sin 2q .
The stress tensor
The stress state at a point is represented mathematically by a second-rank tensor. For the tensor, the values of the stress components are taken to be the traction components acting on the negative sides of the infinitesimal cube; this convention removes some of the ambiguity associated with the sign of shear stresses in particular, but may be opposite that of the Mohr circle convention (as discussed above).
There always exists a coordinate system in which the shear stresses on all three principle planes are simultaneously zero and the normal stresses extrema. In order to analyze a given system using only 2-D notation, the third (unconsidered) dimension must be parallel to the intermediate principle stress.
Certain additional conventions are important to keep in mind when diagramming the physical state of a system:
To avoid error when analyzing the stress state of a system, one should keep detailed records, such as tables of component values, noting which convention is being followed for a particular part of the analysis.
Three-dimensional analysis: the extended Mohr circle
3-D stress plots on a Mohr diagram as three interrelated circles. The outer one is simply that for 2-D stress, with maximum and minimum values along the normal stress axis of s 1 and s 3, respectively. s 2, the intermediate principle stress lies on the normal stress axis between these two where the two lesser circles (coincidentally) cross. For a plane not parallel to one of the principle axes, the components of stress on that surface must lie inside the s 1-s 3 circle and outside the two inner circles. Shear stress maxima for planes of all possible orientations occur only on the s 1-s 3 circle at 2q 2 = 90° .
Specific states of stress
Certain stress states are so distinctive or important that they are given names. Hydrostatic pressure (and lithostatic pressure) is a state in which all principle stresses are equal and compressive (s 1 = s 2 = s 3 = p). There are two possibilities for uniaxial stress: uniaxial compression, where s 1 > s 2 = s 3 = 0, and uniaxial tension, where 0 = s 1 = s 2 > s 3. In axial (or confined) compression or extension, all three principle stresses are positive, but either the greater two (compression) or lesser two (extension) are equal. Triaxial stress is the most general case, where all three principle stresses are unequal.
In pure shear stress s 1 = -s 3 and s 2 = 0. Deviatoric stress, which is important in analyzing (for example) metamorphism, is represented as a tensor D s kl, and is obtained by subtracting the mean normal stress from each component of the stress tensor when k = l. Differential stress Ds is simply the difference between the maximum and minimum principles stresses (s 1 s 3).