Concept of Stress and Strain

Stress

* Stress is the **force per unit area** acting on a rock body.
* Formula:

sigma ={F}/{A}
 

  where $F$ = force, $A$ = area.
* Types of stress:

  1. **Normal stress ($\sigma_n$)** – acts perpendicular to a surface.
  2. **Shear stress ($\sigma_s$)** – acts parallel to a surface.
* Stress is the cause, and strain is the effect.

### **Strain**

* Strain is the **change in shape or size** of a rock body due to applied stress.
* It is **dimensionless** (ratio of change/original).
* Types of strain:

  1. **Elastic strain** – temporary, rock returns to original shape.
  2. **Plastic strain** – permanent deformation without fracture.
  3. **Brittle strain** – rock breaks after exceeding strength.

---

## **Strain Ellipse & Strain Ellipsoid**

Strain in rocks is often represented geometrically by ellipses (2D) and ellipsoids (3D).

### **Strain Ellipse (2D)**

* In 2D deformation, a circle of originally undeformed particles becomes an **ellipse** after strain.
* Axes of the strain ellipse:

  * **$X$** = maximum extension axis
  * **$Y$** = intermediate axis
  * **$Z$** = maximum shortening axis (in 2D only two axes are visible).

#### **Types of Strain Ellipses (based on deformation)**

1. **Pure shear ellipse (coaxial strain):**

   * Symmetrical stretching & shortening.
   * Axes remain fixed.
2. **Simple shear ellipse (non-coaxial strain):**

   * Shear dominates, ellipse rotates with strain.
3. **General strain ellipse:**

   * Combination of pure and simple shear.

---

### **Strain Ellipsoid (3D)**

* In 3D deformation, a sphere of undeformed material becomes an **ellipsoid**.
* Axes of the strain ellipsoid:

  * **$X$ (long axis):** maximum extension
  * **$Y$ (intermediate axis):** medium strain
  * **$Z$ (short axis):** maximum shortening

#### **Types of Strain Ellipsoids**

1. **Prolate ellipsoid (cigar-shaped):**

   * $X > Y \approx Z$
   * Formed by stretching in one direction.
   * Example: boudinage structures.

2. **Oblate ellipsoid (pancake-shaped):**

   * $X \approx Y > Z$
   * Formed by flattening.
   * Example: cleavage development in slates.

3. **Triaxial ellipsoid (general case):**

   * $X > Y > Z$
   * Unequal strain in all directions.
   * Example: complex folding zones.

---

## **Properties of Strain Ellipses & Ellipsoids**

* They represent **finite strain** (total deformation, not instantaneous).
* Axes indicate **principal strain directions**.
* Geometry helps distinguish between **flattening** vs **stretching** strain.
* They can be reconstructed from deformed fossils, pebbles, or reduction spots.

---

## **Geological Significance**

1. **Strain analysis:** Used to determine deformation history of rocks.
2. **Tectonic interpretation:** Helps understand regional stress fields (compression, extension, shear zones).
3. **Structural evolution:** Explains folding, cleavage, boudinage, lineation, and foliation development.
4. **Restoration of original shapes:** Important for paleogeographic reconstructions (e.g., restoring fossils or pebbles).
5. **Shear sense indicators:** Rotated strain ellipses in shear zones help determine movement direction.