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Stress II: Stress Tensors & Mohr Circle Representation
Ch. 4, p. 72-‐
1. Stress at a Point: Because the state of stress is different on every plane passing through a point, one plane cannot be used to understand the stress field. In order to fully characterize stress in a rock volume, we need to consider the stresses acting across all possible plane orientations passing through that point.
[Figure. 2D representation of 5 surfaces passing through a point. In actuality, there are an infinite number of possible planes]
2. Stress at a Point: As we cannot consider an infinite number of planes, we still need to have some way of thinking about how stress acts on a point. Any one plane will have two tractions defining the surface stress that resolves onto that plane from the surrounding stress field.
[Figure. Tractions that resolve onto a single plane in 2D]
3. Stress at a Point: If we consider all tractions for all possible planes, and scale them according to their relative magnitudes, they trace out the shape of an ellipse in 2D and an ellipsoid in 3D. These represent the stress ellipse (2D) and stress ellipsoid (3D).
_Q: what is the minimum amount of information needed to fully define (a) an ellipse and (b) an ellipsoid? (a) _________________ (b) __________________
[Fig. 4.3. Stress ellipse in 2D and stress ellipsoid in 3D]
4. Stress at a Point: Three mutually orthogonal axis lengths are needed to fully characterize an ellipsoid. These are the maximum, intermediate, and minimum principal stresses, respectively (σ 1 , σ 2 , and σ 3 ). So σ 1 > σ 2 > σ 3 always.
The principal stresses are simply three examples of surface stresses out of the infinite number that actually exist. They specifically act on three planes in such a way that they are perpendicular to those planes (i.e., σ 1 , σ 2 , and σ 3 are normal stresses). These are principal planes and they must contain zero shear stress.
[Fig. 4.3. Stress ellipsoid in 3D, showing principal stresses and principal planes]
5. Stress at a Point: Although 3 principal stresses can be used to characterize stress at a point, we can also use the concept of the stress tensor, which allows us to characterize stresses in a common coordinate system or frame of reference.
We do this by approximating the point as an infinitesimally small cube and considering the stress components (normal and shear) that act on the faces of this cube. The cube is oriented such that each of its faces has a normal vector that points along one of the defined coordinate axes (x, y, or z).
[Fig. 4.5. Concept of the stress cube in 3D to approximate the state of stress at a point]
6. Stress at a Point: If the principal stresses happened to act in the x,y,z directions, they would be acting normal to the faces of the cube. The faces of the cube would then be principal planes and there would be no shear stresses on them.
In general, principal stresses may be oriented at some angle to our chosen coordinate system (which may represent N-‐S, E-‐W, and up-‐down). In this case, each face of the cube experiences both normal stress and shear stress.
[Fig. 4.5. Concept of the stress cube in 3D to approximate the state of stress at a point]
7. Stress at a Point: In fact, each face would experience one normal stress and two shear stresses.
Each component is labeled with a double subscript notation (e.g., σxy ), which respectively represent the direction of the normal to the surface upon which they act, followed by the direction in which they act. This is called the on-‐in notation.
8. Stress at a Point: As this is an infinitesimally small cube, we assume the tractions on one face have equipollent tractions on the opposite face. There are thus 3 sets of faces, each with 3 stress components for a total of 9 stress components. They are:
σxx , σyy , σzz : 3 normal stresses σxy , σyx , σxz , σzx , σyz , σzy : 6 shear stresses
These 9 components fully define the state of stress at a point and indicate that stress is a tensor quantity. A tensor requires magnitudes, directions, and orientation of planes upon which the components act (unlike vectors).
[Fig. 4.5. Concept of the stress cube in 3D to approximate the state of stress at a point]
9. Stress at a Point: For a stress cube at rest (i.e., not accelerating or rotating), we must also have a balance of stresses to allow equilibrium. This requires that:
σxy = σyx σxz = σzx σyz = σzy
There are thus only 6 independent components to any stress tensor.
[Fig. 4.5. Concept of the stress cube in 3D to approximate the state of stress at a point]
10. Stress Tensor Matrix: The 9 components of a stress tensor are represented in the form of a 3 x 3 matrix, arranged in a specific order:
" xx " xy " xz
" yx " yy " yz
" zx " zy " zz
The normal stresses define a diagonal from top left to bottom right ( circle these on the above matrix ).
17. Representing Stress on a Mohr Diagram: Note that angle θ in physical space is always doubled to 2θ in Mohr space. So the full range of plane orientations is 360° in Mohr space (hence, the circle). The angle 2θ is always measured in a CCW sense away from the right side of the circle, along the σn axis. 18. Representing Stress on a Mohr Diagram: The Mohr circle shows us that any plane (red star in figure) will have two complementary planes that respectively contain the same amount of shear stress but a different normal stress (orange star), or the same normal stress but an opposite sign of shear stress (yellow star).
[Fig. 4.6. Mohr diagram representation of stress]
19. Constructing a Mohr Circle: By definition, any surface containing zero shear stress (no sliding possible) has a normal stress that is also a principal stress (i.e., a principal plane). Therefore, principal stresses must plot along the σn axis, where σs = 0. For the 2D case, we only plot σ 1 and σ 3.
Because σ 1 > σ 3 in magnitude, it plots further to the right on the σn axis.
[Figure. Mohr diagram axes]
20. Constructing a Mohr Circle: We typically only represent +ive shear and normal stresses, so a semi-‐circle is drawn that connects σ 1 and σ 3. Given the definition of θ, if σ 1 is perpendicular to a plane, θ = 0° (i.e., 2θ = 0°). This is why 2θ is measured from the right side of the circle, where 2θ = 0°.
In physical space, the range of θ between σ 1 and σ 3 is 90°, so in Mohr space this angle must be 180°. This allows both σ 1 and σ 3 to plot on the σn axis, which is necessary as they are normal stresses to principal planes.
[Figure. Constructing a Mohr diagram]
21. Using a Mohr Circle: Read off the coordinates of (σn , σs ) along the circle for any value of 2θ to determine the normal and shear stress along any plane at angle θ to σ 3.
[Figure. Using a Mohr circle diagram]
