Stress diagram

In the structural geology, students get difficulties on the concept of stress. First of all, we will understand what is stress. The terms pressure and stress are often used interchangeably. In geology, Pressure (p) is generally limited to media with no or very low shear resistance (fluids), while stress (s) is used when dealing with media with a minimum of shear resistance (rocks). There are two different types of forces. One type affects the entire volume of a rock, the outside as well as the inside, and is known as body forces. Body forces define three dimensional fields. The most important type of body force in structural geology is gravity. Another example is magnetic forces. The other type of force acts on surfaces only and is referred to as surface forces. Surface forces originate when one body pushes or pulls another body. The force that acts across the contact area between the two bodies is a surface force. Surface forces are of great importance during deformation of rocks. In a similar way we can talk about stress on a surface and state of stress at a point. Stress on a plane is a vector quantity, while stress at a point is a second-order tensor. In this post you will understand the concept of principal stresses and principal planes.

Stress on a surface

The stress on a surface such as a fracture or a grain–grain contact is a vector (s) that can be defined as the ratio between a force (F) and the area (A) across which the force acts. Compressive stresses are normally considered positive in the geologic literature, while tension is regarded as negative.

Stress on a surface diagram

Stresses in the lithosphere are almost everywhere compressional, even in rifts and other areas undergoing extension

Concept of Normal stress and shear stress in structural geology

A stress vector oriented perpendicular to a surface is called the normal stress on that surface, while a stress vector that acts parallel to a surface is referred to as the shear stress. In general, stress vectors act obliquely on planes. The stress vector can then be resolved into normal and shear stress components. It is emphasized that the concept of normal and shear stress has a meaning only when related to a specific surface. This diagram will guide you to understand Normal stress and shear stress. Check also the formula of Normal stress and shear stress.

Stress diagram

Normal stress and shear stress formula

Normal and shear stress formula

Stress at a point

We may imagine that there are planes in an infinite number of orientations through this point. Perpendicularly across each of the planes there are two oppositely directed and equally long traction or stress vectors. Different pairs of stress vectors may be of different lengths, and when a representative family of such vectors is drawn about the point an ellipse emerges in two dimensions (see the previous figure) An obvious requirement for expressing stress as an ellipse (ellipsoid) is that there is not a combination of positive and negative tractions. The ellipse is called the stress ellipse, and the ellipsoid is the stress ellipsoid.

The stress ellipsoid and its orientation tell us everything about the state of stress at a given point in a rock, or in a rock volume in which stress is homogeneous

The stress ellipsoid has three axes, denoted σ1, σ2 and σ3. The longest (σ1) is the direction of maximum stress while the shortest is normal to the (imaginary) plane across which there is less traction than across any other plane through the point. The axes are called the principal stresses and are the poles to the principal planes of stress. These are the only planes where the shear stress is zero.

Stress components

The state of stress at a point is also defined by the stress components that act on each of the three orthogonal surfaces in an infinitesimal cube. Each of the surfaces has a normal stress vector (sn) and a shear stress vector (ss) along each of its two edges, as illustrated in Figure below. In total, this gives three normal stress vectors and six shear stress vectors. If the cube is at rest and stable the forces that act in opposite directions are of equal magnitude and hence cancel each other out. This implies that

Stress components

 

σxy = – σyx , σyz = -σzy

and

σxz= -σzx

and we are left with six independent stress components. The cube can be oriented so that all of the shear stresses are zero, in which case the only non-zero components are the three normal stress vectors. In this situation these vectors represent the principal stress directions and are the principal stresses or principal axes of the stress ellipsoid. The three surfaces that define the cube are the principal planes of stress that divide the stress ellipsoid into three.

Principal stresses and planes

  1. A body or a plane may subjected to only normal stress (either tensile or compressive stress)

Normal stress

  1. A body or a plane may be subjected to only shear stress or tangential stress.Shear stress
  2. A body or a plane may even be subjected to a combination of normal stress as well as tangential stress (shear stress)Normal and shear stress

Now, the tensile or compressive stresses given in the problem would be denoted by σx or σy or vice versa.

Here, is detailed discussion on the two dimensions stresses

If we consider an internal plane inclined at some angle ‘θ’ to the direction of loading, it is clear that both normal and tangential forces will exist on the plane to maintain equilibrium. Therefore a system of direct and shear stresses will exist on the inclined plane.

Consider the bar shown in Figure 1 subjected to an axial force F through the centroid of the cross section. A rectangular element ABCD within the bar will be subjected to direct stresses (Force/Area) = F/(AD)t on faces AD and BC of the element.

Normal stress one dimension

Consider now the stresses on plane AN, inclined at angle f to AD. For equilibrium of the triangular element to be maintained, a direct stress sf and a shear stress tf will be required to act on the plane.

Stress Transformation

The Stress Transformation equations can be used to calculate the normal and shear stresses on any such plane through a point where the state of stress is defined. The stresses on this inclined plane can be determined by a graphical construction.

reference:

Structural Geology by Haakon Fossen

 

 

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