What is volumetric strain?

Short Answer:

Volumetric strain is the type of strain that occurs when there is a change in the total volume of a material due to applied stress or pressure. It is defined as the ratio of the change in volume (ΔV) to the original volume (V) of the material.

In simple words, volumetric strain measures how much the overall size or volume of a body increases or decreases when subjected to uniform stress in all directions. It has no unit, as it is a ratio, and is commonly used in cases where materials are compressed or expanded, such as in pressure vessels or fluid containers.

Detailed Explanation:

Volumetric Strain

Definition and Meaning:
Volumetric strain is defined as the ratio of the change in volume of a body to its original volume when the body is subjected to external pressure or stress acting equally in all directions.

Mathematically,

Where,

  • ΔV = Change in volume
  • V = Original volume

Volumetric strain is a dimensionless quantity because it is the ratio of two quantities having the same unit (volume). It is an important concept in the study of stress and strain, especially in materials that are exposed to hydrostatic pressure or three-dimensional stress conditions.

In simple terms, when a material is compressed from all sides, its volume decreases. Similarly, when it is expanded due to internal pressure, its volume increases. The extent of this change in volume, compared to its original volume, is measured by volumetric strain.

Explanation of Volumetric Strain

When a body is subjected to uniform stress from all sides — for example, when placed inside a pressurized container or submerged in a fluid — it experiences a uniform compression or expansion.
In such cases, every dimension of the material (length, breadth, and height) changes slightly, and the overall volume of the body either increases or decreases.

The ratio of this total volume change to the initial volume gives the volumetric strain. It represents the three-dimensional deformation of the material rather than a linear one.

Volumetric strain is not limited to solids. It also applies to fluids and gases, which easily change their volume under pressure. For example, when air inside a piston is compressed, its volume decreases, leading to an increase in volumetric strain.

Derivation of Volumetric Strain in a Cube or Rectangular Block

Let us consider a cube of side L subjected to equal stresses (σ) on all faces.
Due to this stress, each dimension (length, breadth, and height) will change slightly.

If the change in length due to stress in one direction is ΔL, the linear strain in that direction is:

Since the stress acts equally in all three directions, the total change in volume of the cube will be the sum of the changes along the three directions (x, y, and z).

Therefore,

If the material is isotropic (same properties in all directions), then all three strains are equal, and

However, in a general case where the material is under three different normal stresses (σx, σy, σz), the total volumetric strain is given by:

where,

  • E = Young’s modulus
  • μ = Poisson’s ratio

This equation helps engineers calculate volumetric deformation in complex stress conditions.

Significance of Volumetric Strain

  1. Design of Pressure Vessels and Tanks:
    In industries, vessels and tanks that hold gases or liquids are designed to withstand internal and external pressures. The concept of volumetric strain helps determine how much these containers expand or contract under pressure.
  2. Study of Compressibility:
    The property of a material to reduce its volume under pressure is called compressibility. Volumetric strain helps calculate the compressibility of materials like gases, liquids, and metals.
  3. Structural Analysis:
    Engineers use volumetric strain to understand how solid structures behave under hydrostatic stress, especially in deep-sea, underground, or high-pressure environments.
  4. Fluid Mechanics and Thermodynamics:
    Volumetric strain is closely related to the behavior of fluids and gases under varying pressure and temperature, forming the basis of laws like Boyle’s Law and Bulk Modulus calculations.

Relation between Volumetric Strain and Bulk Modulus

The Bulk Modulus (K) is a property that defines how resistant a material is to uniform compression. It is given by the ratio of the applied stress to the resulting volumetric strain.

Mathematically,

or

Here,

  • A high bulk modulus means the material is difficult to compress (e.g., steel).
  • A low bulk modulus means the material is easily compressible (e.g., rubber, gases).

This relationship helps engineers understand how materials react under pressure, which is crucial in designing pressure systems, hydraulic machines, and submarine structures.

Examples of Volumetric Strain

  1. Compressed Gas Cylinder:
    When gas is filled under high pressure, the cylinder slightly expands due to internal pressure. The ratio of this expansion to its original volume gives the volumetric strain.
  2. Underwater Structures:
    Deep-sea structures experience high external water pressure, causing their volume to decrease — a case of compressive volumetric strain.
  3. Hydraulic Systems:
    The oil inside hydraulic systems experiences a change in volume under pressure, which must be considered in the system’s design.
  4. Earth’s Crust:
    Rocks and minerals deep under the earth’s surface experience volumetric strain due to high natural pressure.

Factors Affecting Volumetric Strain

  1. Magnitude of Applied Pressure: Greater pressure results in higher volumetric strain.
  2. Material Properties: Materials with higher bulk modulus experience less strain.
  3. Temperature: Increase in temperature may expand materials, affecting volume change.
  4. Type of Stress: Only uniform or hydrostatic stress causes true volumetric strain.
  5. Poisson’s Ratio: Higher Poisson’s ratio means more lateral deformation, influencing volumetric strain.
Conclusion:

Volumetric strain is the ratio of the change in volume to the original volume of a material when it is subjected to uniform stress or pressure. It measures the overall deformation of a material in three dimensions rather than just a single direction. Volumetric strain is a key parameter in understanding how materials behave under pressure and is closely linked to bulk modulus and compressibility. It is essential in designing and analyzing components like pressure vessels, hydraulic systems, and structural parts exposed to uniform stress in all directions.