Mastering True Strain: A Deep Dive into Material Deformation

When studying the mechanics of materials, one of the most critical concepts for engineers to grasp is how objects change shape under stress. While many are introduced to Engineering Strain early in their studies, professional structural analysis and manufacturing simulations often require the use of True Strain (also known as logarithmic or natural strain). This guide explores the definition, formula, and practical significance of true strain in modern physics and engineering.

What is True Strain?

True strain is a measure of deformation that accounts for the instantaneous change in length of a specimen during a loading process. Unlike engineering strain, which references the original dimensions of the object, true strain integrates the incremental changes in length over the entire deformation path. This makes it a far more accurate representation of material behavior, especially in scenarios involving large deformations or plastic flow, such as metal forging or deep drawing.

The Mathematical Formula

The fundamental formula for True Strain (ε) is derived from the integration of incremental changes in length (dL) relative to the current length (L):

Where:

• ε (epsilon): True Strain (dimensionless).
• L: Instantaneous or Final Length.
• L₀: Original or Initial Length.
• ln: The Natural Logarithm.

True Strain vs. Engineering Strain

The difference between these two measurements is negligible at low deformation levels (typically less than 1%). However, as a material is stretched significantly, the values diverge. Engineering strain (e) is defined as:

A useful relationship between the two for small deformations is: ε = ln(1 + e). This relationship holds true until the material begins to “neck” or experience localized thinning, at which point the volumetric changes and cross-sectional area changes make true strain the only reliable metric.

Why Engineers Use True Strain

1. Additivity: One of the greatest mathematical advantages of true strain is that it is additive. If a material is stretched from L₀ to L₁ and then from L₁ to L₂, the total true strain is simply the sum of the two individual steps. Engineering strain does not possess this property.
2. Volume Conservation: In plastic deformation of metals, the volume remains essentially constant. True strain accounts for the simultaneous reduction in cross-sectional area as the length increases, providing a consistent view of the material’s state.
3. Accuracy in Large Strains: In manufacturing processes like extrusion or wire drawing, materials undergo massive shape changes. Using engineering strain in these contexts would lead to massive errors in calculating the required force and energy.

Practical Example

Imagine a metal rod with an initial length of 100 mm. After being pulled in a tensile test, its length increases to 150 mm. Let’s calculate both values:

• Engineering Strain: (150 – 100) / 100 = 0.50 (or 50%).
• True Strain: ln(150 / 100) = ln(1.5) ≈ 0.4055.

As you can see, the engineering strain overestimates the “true” deformation because it continues to reference the original 100 mm length rather than the increasing length of the rod as it stretches.

True Stress-Strain Curves

In materials science, plotting a True Stress vs. True Strain curve is essential for determining the work-hardening exponent. Unlike the engineering stress-strain curve—which appears to drop after the ultimate tensile strength point due to “necking”—the true stress-strain curve continues to rise, showing that the material is actually getting stronger as it deforms (strain hardening) right up until fracture.

Summary Table

Feature	Engineering Strain	True Strain
Reference Length	Original (L₀)	Instantaneous (L)
Calculation	ΔL / L₀	ln(L / L₀)
Additive?	No	Yes

Frequently Asked Questions