Short Answer:

True stress is the ratio of the instantaneous load to the actual cross-sectional area of a material at any given moment during deformation. It considers the continuous change in the area as the material stretches or compresses.

True strain is the natural logarithm of the ratio of the instantaneous length to the original length of a material. It represents the continuous accumulation of small changes in length during deformation. Both true stress and true strain give a more accurate representation of material behavior, especially beyond the elastic limit.

Detailed Explanation:

True Stress and True Strain

Definition and Meaning:
When a material is subjected to a gradually increasing load, it undergoes deformation. Initially, the relationship between stress and strain is linear, and the material behaves elastically. However, as the load continues to increase, the material begins to deform plastically, and its cross-sectional area and length change significantly.

In such cases, nominal (engineering) stress and strain do not accurately represent the actual condition of the material, because they assume the original area and length remain constant. Therefore, true stress and true strain are used to express the actual behavior of a material during deformation.

1. True Stress

Definition:
True stress is defined as the ratio of the instantaneous load acting on the material to the instantaneous (actual) cross-sectional area at that particular moment during deformation.

Mathematically,

Where:

• σt = True stress (N/m² or Pa)
• P = Applied load (N)
• At = Instantaneous cross-sectional area (m²)

Unlike engineering stress, which uses the original area (A₀), true stress accounts for the continuous reduction in area as the material elongates under tension.

Explanation:
When a specimen is stretched, its length increases and its cross-sectional area decreases. Since the load continues to act on a smaller area, the true stress increases more rapidly than the engineering stress.

Thus, true stress gives a more precise indication of the internal resistance offered by the material during plastic deformation.

2. True Strain

Definition:
True strain is defined as the natural logarithm (logₑ) of the ratio of the instantaneous length to the original length of the specimen.

Mathematically,

Where:

• εt = True strain (dimensionless)
• Lt = Instantaneous length
• L₀ = Original length

Explanation:
In a real situation, deformation occurs gradually, not all at once. True strain considers the sum of all infinitesimal increments of strain during the deformation process. Therefore, it gives a more realistic measure of the actual deformation that the material undergoes.

In contrast, engineering strain assumes deformation happens uniformly and instantly, which is valid only for small strains within the elastic range.

Relation Between Engineering and True Values

Since both engineering and true stress-strain are used in practice, the relationship between them is very useful for conversion.

1. Relation for Stress:

where,

1. • σt = True stress
   • σ = Engineering stress
   • ε = Engineering strain
2. Relation for Strain:

where,

1. • εt = True strain
   • ε = Engineering strain

These relationships are valid up to the point of uniform deformation, i.e., before necking starts in a tensile test specimen.

Difference Between True and Engineering Values

Aspect	Engineering Stress/Strain	True Stress/Strain
Area and Length Used	Original area and length	Instantaneous area and length
Accuracy	Approximate for large deformations	Accurate throughout deformation
Applicable Range	Valid up to elastic limit	Valid beyond yield point and into plastic region
Mathematical Simplicity	Easier to calculate	Requires continuous measurement
Representation	Linear till yield	Non-linear and more realistic curve

Graphical Representation

In a stress-strain diagram:

• The engineering stress-strain curve rises linearly up to the elastic limit, then shows yielding, ultimate stress, and finally a drop during necking.
• The true stress-strain curve, however, continues to rise even after necking because it accounts for the reduced cross-section area and true deformation.

Hence, the true curve gives a better representation of the material’s actual behavior, especially in plastic deformation and failure regions.

Importance of True Stress and True Strain

1. Accurate Material Behavior:
   True values provide realistic results because they consider changes in both area and length during deformation.
2. Useful in Plastic Deformation Studies:
   They are essential for analyzing the behavior of materials beyond the yield point, especially during processes like forging, rolling, or extrusion.
3. Design of Forming Processes:
   In metal forming, the materials experience large strains and area reduction. True stress and strain are used to calculate the forces required for shaping operations.
4. Prediction of Failure:
   True stress helps predict when and how a material will fail, providing more reliable data for safety designs.
5. Scientific and Industrial Applications:
   They are widely used in advanced mechanical analysis, computer simulations, and finite element methods for accurate prediction of deformation.

Example Calculation

Let:
Original length,
Instantaneous length,
Original area,
Instantaneous area,
Applied load,

Then,

 

Thus,
True Stress = 100 MPa,
True Strain = 0.182.

This demonstrates that true values are higher than corresponding engineering values because they account for continuous changes in geometry.

Applications in Engineering

1. Metal Forming Operations: Used in forging, rolling, extrusion, and drawing processes.
2. Finite Element Analysis: For accurate simulation of material deformation.
3. Tensile and Compression Testing: To determine true yield strength and ductility.
4. Design of Plastic Components: Helps predict long-term deformation behavior.
5. Fracture Mechanics: Used in studying material failure beyond elastic limits.

Conclusion:

True stress is the ratio of the applied load to the actual area at any instant during deformation, while true strain is the natural logarithm of the ratio of the current length to the original length. Both are more accurate than engineering values, especially when the material undergoes large deformations beyond the elastic range. True stress and true strain provide a realistic representation of a material’s strength and ductility, which is essential in modern material design, testing, and metal forming applications.