formula for true strain

2 min read 19-10-2024

Unraveling the True Strain: A Deep Dive into the Formula

The true strain, also known as logarithmic strain, is a crucial concept in material science, particularly when dealing with large deformations. Unlike the engineering strain, which is based on the original dimensions, the true strain considers the changing dimensions of a material as it deforms. This makes it a more accurate measure of the material's behavior under stress.

Understanding the Formula

The formula for true strain is:

ε_true = ln(l/l₀)

Where:

ε_true is the true strain
ln is the natural logarithm
l is the current length of the material
l₀ is the original length of the material

Why is True Strain Important?

Accurate Representation of Deformation: True strain accounts for the changing geometry of the material during deformation, making it a more accurate reflection of the material's actual behavior.
Consistency with Material Properties: Many material properties, like the yield strength, are defined based on true stress and true strain, ensuring a consistent and reliable understanding of the material's response.
Application in Complex Scenarios: True strain is particularly useful in situations involving large plastic deformations, where the engineering strain becomes unreliable due to the significant changes in the material's geometry.

Let's Illustrate with an Example:

Imagine a rubber band with an initial length of 10 cm. We stretch it to a length of 15 cm.

Engineering Strain: (15 cm - 10 cm) / 10 cm = 0.5
True Strain: ln(15 cm / 10 cm) = ln(1.5) ≈ 0.405

Notice how the true strain is smaller than the engineering strain. This is because the true strain accounts for the fact that the rubber band is constantly stretching and changing its length as we apply the force.

Relationship between True Strain and Engineering Strain

For small deformations, the true strain and engineering strain are roughly equivalent. However, as the deformation becomes larger, the difference between them becomes more significant.

The relationship between the two can be expressed as:

ε_true = ln(1 + ε_engineering)

This formula can be used to convert between the two strain measures depending on the context of the problem.

Key Takeaways:

The true strain is a more accurate measure of deformation, especially for large strains.
It is essential for understanding material behavior, particularly in complex deformation scenarios.
It is directly related to the true stress and is used in defining important material properties.

By understanding the concept of true strain and its formula, engineers and scientists can better analyze and predict the behavior of materials under stress, leading to safer and more efficient designs.