Understanding the True Stress Formula for Plastic Deformation Studies
In the realm of mechanical engineering and materials science, accurately characterizing a material's response to applied forces is paramount. While engineering stress and strain are widely used, they can be misleading, especially when dealing with significant plastic deformation. This is where the concept of true stress and true strain becomes crucial. The true stress formula provides a more accurate representation of the stress experienced by a material during deformation, particularly when the cross-sectional area changes substantially. This article will delve into the true stress formula, its derivation, application in plastic deformation studies, and comparisons with engineering stress, along with practical examples to solidify understanding.
The formula for true stress, often denoted as σt, is given by:
σt = F/Ai
Where: σt is the true stress
F is the instantaneous applied force
Ai is the instantaneous cross-sectional area of the material at that specific point in time during deformation.
The Need for True Stress
Engineering stress (σe), calculated as the applied force divided by theoriginalcross-sectional area (Ao), works well for small elastic deformations. However, as a material undergoes plastic deformation, its cross-sectional area changes. This change becomes significant, especially during processes like tensile testing, necking, and metal forming operations. Engineering stress does not account for this area reduction. Therefore, it can give a false representation of the material's resistance to deformation at higher strains. True stress, on the other hand, considers the instantaneous area, providing a more accurate measure of the force acting on the reduced cross-section.
Derivation of the True Stress Formula (from Engineering Stress)
The relationship between true stress and engineering stress can be derived based on the assumption of constant volume during plastic deformation (a reasonable assumption for many metals). This means:
AoLo = AiLi
Where:
Ao is the original cross-sectional area
Lo is the original length
Ai is the instantaneous cross-sectional area
Li is the instantaneous length
From this, we can express the ratio of the areas as:
Ao/Ai = Li/Lo
Recall that engineering strain (εe) is defined as:
εe = (Li - Lo)/Lo = (Li/Lo) - 1
Therefore:
Li/Lo = εe + 1
Substituting this into the area ratio:
Ao/Ai = εe + 1
Now, we know engineering stress is σe = F/Ao and true stress is σt = F/Ai. Dividing the true stress equation by the engineering stress equation:
σt/σe = (F/Ai) / (F/Ao) = Ao/Ai = εe + 1
Therefore, the relationship between true stress and engineering stress is:
σt = σe (1 + εe)
The true strain (εt) can also be related to the engineering strain as:
εt = ln(1 + εe)
Therefore, you can also express the true stress as a function of true strain if you substitute appropriately.
Application in Plastic Deformation Studies
The true stress-strain curve provides vital information for understanding and predicting material behavior during plastic deformation. This is particularly important in several engineering applications: Metal Forming: Processes like forging, rolling, and extrusion involve significant plastic deformation. True stress-strain curves are essential for designing dies and predicting the forces required for these operations. Structural Analysis: When structures are subjected to extreme loads that cause plastic deformation, analyzing the true stress distribution is crucial for ensuring structural integrity and preventing catastrophic failure. Finite Element Analysis (FEA): Many FEA software packages require true stress-strain data as input to accurately simulate material behavior under large deformations. Material Characterization: Comparing true stress-strain curves for different materials allows engineers to select the best material for a specific application where plastic deformation is expected.
Example Calculation
Let's consider a cylindrical steel rod with an initial diameter of 10 mm. It is subjected to a tensile force. At a certain point during the test, the applied force is 30,000 N, and the diameter has reduced to 8 mm. Calculate the engineering stress and the true stress.
1. Engineering Stress
Original area, Ao = π (10 mm/2)2 = 78.54 mm2
Engineering stress, σe = F/Ao = 30,000 N / 78.54 mm2 =
381.97 N/mm2 (or MPa)
2. True Stress
Instantaneous area, Ai = π (8 mm/2)2 = 50.27 mm2
True stress, σt = F/Ai = 30,000 N / 50.27 mm2 =
596.87 N/mm2 (or MPa)
As you can see, the true stress (596.87 MPa) is significantly higher than the engineering stress (381.97 MPa), highlighting the importance of using true stress when dealing with large plastic deformations.
Example 2: Using Engineering Strain to Find True Stress
Consider a copper wire with an original length of 50 mm and an original cross-sectional area of 2 mm2. The wire is stretched to a length of 55 mm under a tensile force of 200 N. Calculate the engineering stress, engineering strain, and true stress.
1. Engineering Stress
Engineering stress, σe = F/Ao = 200 N / 2 mm2 = 100 MPa
2. Engineering Strain
Engineering strain, εe = (Li - Lo)/Lo = (55 mm - 50 mm)/50 mm = 0.1
3. True Stress
True stress, σt = σe (1 + εe) = 100 MPa (1 + 0.1) = 110 MPa
Comparing True Stress and Engineering Stress
The difference between true stress and engineering stress becomes more pronounced as the deformation increases. For small deformations, the two values are relatively close. However, beyond the elastic limit, the difference widens significantly. Engineering stress typically reaches a maximum value (the ultimate tensile strength) and then decreases as necking occurs. This decrease is an artifact of using the original area in the calculation. In reality, the material is still withstanding increasing stress at the necked region. True stress, however, continues to increase with increasing strain, providing a more accurate representation of the material's behavior.
Common Pitfalls and Misconceptions
Assuming Constant Volume: While the constant volume assumption is generally valid for metals, it may not hold for all materials, especially polymers and composites. In such cases, more complex models may be required. Confusing True Stress with Principal Stress: True stress represents the stress acting on the instantaneous cross-sectional area during uniaxial loading. Principal stresses, on the other hand, represent the maximum and minimum normal stresses at a point under multiaxial loading conditions, regardless of deformation. These are distinct concepts. Using Engineering Stress in FEA for Large Deformations:Employing engineering stress-strain data in FEA simulations involving significant plastic deformation can lead to inaccurate results. It's crucial to use true stress-strain data to ensure the simulation accurately reflects the material's behavior.
People Also Ask
What is the difference between true stress and engineering stress?
Engineering stress is calculated using the original cross-sectional area of a material, while true stress uses the instantaneous cross-sectional area, which changes during deformation. True stress provides a more accurate representation of the stress experienced by the material during plastic deformation, especially when there is a significant reduction in area, such as during necking in tensile testing.
How do you calculate true strain?
True strain (εt) can be calculated using the formula εt = ln(1 + εe), where εe is the engineering strain. Alternatively, it can be expressed as εt = ln(Li/Lo), where Li is the instantaneous length and Lo is the original length.
When is it essential to use true stress instead of engineering stress?
It is essential to use true stress when analyzing materials undergoing significant plastic deformation, such as in metal forming operations (forging, rolling, extrusion), structural analysis under extreme loads causing plastic deformation, and finite element analysis (FEA) simulations where large deformations are expected. Using engineering stress in these scenarios can lead to inaccurate results and potentially unsafe designs.
Conclusion
The true stress formula is a powerful tool for understanding and predicting material behavior during plastic deformation. By accounting for the changes in cross-sectional area, it provides a more accurate representation of the stress experienced by a material than engineering stress. This is crucial for designing reliable and safe structures and processes involving significant plastic deformation. By understanding the derivation, application, and limitations of the true stress formula, engineers can make informed decisions and avoid potential pitfalls in their analyses and designs.