True Stress Formula vs. Nominal Stress Formula: A Comprehensive Guide for Engineers
In the field of mechanical engineering and materials science, understanding stress is paramount to ensuring the safety and reliability of structures and components. Stress, simply put, is the force acting per unit area within a material. However, there are different ways to define and calculate stress, leading to the concepts ofnominal stress(also known as engineering stress) andtrue stress. This article delves into the intricacies of these two stress measures, highlighting their differences, applications, and limitations. Understanding when to use each is crucial for accurate analysis and design in various engineering scenarios.
What is Nominal Stress?
Nominal stress, often denoted as σ (sigma), is the stress calculated using the original cross-sectional area of a material before any deformation occurs. It is a straightforward calculation, making it widely used in introductory engineering courses and preliminary design estimations.
The nominal stress formula is:
σ = F / A₀
where: σ is the nominal stress (typically in Pascals (Pa) or pounds per square inch (psi))
F is the applied force (typically in Newtons (N) or pounds (lb))
A₀ is the original cross-sectional area (typically in square meters (m²) or square inches (in²))
This formula assumes that the area remains constant during deformation, which is a reasonable approximation for small deformations, such as those encountered within the elastic region of a material. However, this assumption breaks down when a material undergoes significant plastic deformation, as its cross-sectional area changes considerably.
Example
Consider a steel bar with an original diameter of 20 mm subjected to a tensile force of 50,000 N. The original cross-sectional area is A₀ = π(10 mm)² = 314.16 mm² =
3.1416 x 10⁻⁴ m².
The nominal stress is:
σ = 50,000 N / (3.1416 x 10⁻⁴ m²) =
159.15 MPa
What is True Stress?
True stress, often denoted as σₜ (sigma true), is the stress calculated using theinstantaneousoractualcross-sectional area of the material during deformation. Unlike nominal stress, true stress accounts for the reduction in area that occurs, especially during tensile loading beyond the yield point. This makes it a more accurate representation of the stress experienced by the material at any given point in time during deformation.
The true stress formula is:
σₜ = F / Aᵢ
where: σₜ is the true stress
F is the applied force
Aᵢ is the instantaneous cross-sectional area
The instantaneous area Aᵢ is often difficult to measure directly during a test. However, for ductile materials undergoing uniaxial tension, a relationship between true strain (εₜ) and nominal strain (ε) can be derived, and consequently a relationship between true stress and nominal stress can be found, assuming volume constancy:
A₀L₀ = AᵢLᵢ
where L₀ and Lᵢ are the original and instantaneous lengths, respectively.
Since ε = (Lᵢ - L₀)/L₀, then Lᵢ = L₀(1 + ε). Substituting,
A₀L₀ = AᵢL₀(1 + ε)
Aᵢ = A₀ / (1 + ε)
Substituting this into the true stress equation:
σₜ = F / (A₀ / (1 + ε)) = (F / A₀)(1 + ε)
σₜ = σ(1 + ε)
Example (continuing from the previous one)
Assume that the steel bar from the previous example elongates such that the nominal strain is 0.1 (10% elongation), and the force remains at 50,000 N. Then theinstantaneousarea is calculated using the volume constancy relationship. Since ε =
0.1, then Aᵢ = A₀ / (1 + ε) =
3.1416 x 10⁻⁴ m² /
1.1 =
2.856 x 10⁻⁴ m².
The true stress is:
σₜ = 50,000 N / (2.856 x 10⁻⁴ m²) =
175.07 MPa
Alternatively, we can calculate it using the relationship derived:
σₜ = σ(1 + ε) = 159.15 MPa (1 +
0.1) =
175.07 MPa
Notice that the true stress (175.07 MPa) is significantly higher than the nominal stress (159.15 MPa), demonstrating the effect of the decreasing cross-sectional area.
Difference Between True Stress and Nominal Stress
The key difference lies in the area used for calculation. Nominal stress utilizes theoriginalarea, while true stress uses theinstantaneousarea. This difference becomes significant when the material undergoes substantial plastic deformation.
Here's a table summarizing the key differences:
| Feature | Nominal Stress (Engineering Stress) | True Stress |
|---|---|---|
| ----------------- | --------------------------------------- | ------------------------------------------------- |
| Area Used | Original Cross-Sectional Area (A₀) | Instantaneous Cross-Sectional Area (Aᵢ) |
| Formula | σ = F / A₀ | σₜ = F / Aᵢ = σ(1 + ε) (assuming volume constancy) |
| Applicability | Small Deformations (Elastic Region) | Large Deformations (Plastic Region) |
| Accuracy | Less Accurate for Large Deformations | More Accurate for Large Deformations |
| Calculation Ease | Easier | More Complex |
When to Use True Stress vs. Nominal Stress
Nominal Stress: Suitable for situations involving small deformations, such as within the elastic range of a material. It's often used in preliminary design calculations and when dealing with materials that exhibit little to no plastic deformation before failure. Examples include designing for components that must not exceed their yield strength under normal operating conditions. True Stress: Crucial when analyzing materials undergoing significant plastic deformation, such as in metal forming processes (e.g., forging, extrusion, drawing), or when predicting the behavior of materials near failure. It is also essential for finite element analysis (FEA) where accurate stress representation is needed under large strain conditions.
Real-World Applications
Pressure Vessels: During hydrostatic testing or operation at high pressures, pressure vessels can experience significant deformation. True stress calculations are crucial to accurately assess the vessel's integrity and predict its lifespan, especially when considering strain hardening effects. Beams: While nominal stress is often used for beam bending analysis in the elastic range, true stress may be relevant when analyzing beams undergoing large deflections or plastic hinge formation. Rotating Machinery: Components in rotating machinery, such as turbine blades, can experience complex stress states due to centrifugal forces and thermal gradients. Both nominal and true stress analysis can be important, especially when assessing creep and fatigue failure. Structural Analysis: In structural engineering, understanding stress distribution is essential for ensuring the stability and safety of buildings and bridges. For situations where materials are strained beyond their elastic limit (e.g., during an earthquake), true stress analysis provides a more realistic assessment of structural integrity. Thermal Stress: Thermal expansion and contraction can induce significant stress in constrained materials. If the thermal stress exceeds the yield strength, plastic deformation can occur, requiring true stress analysis for accurate modeling. Metal Forming: Processes like forging, rolling, and extrusion involve large plastic deformations. True stress and true strain are the relevant parameters for designing these processes and predicting the final product's properties.
Common Pitfalls and Misconceptions
Assuming Constant Area: A common mistake is to always use nominal stress, even when dealing with significant plastic deformation. This can lead to underestimation of the actual stress experienced by the material and potentially unsafe designs. Confusion with Yield Strength: The yield strength obtained from a tensile test is typically based on nominal stress. However, the actual stress required to continue plastic deformation (as reflected by the true stress) will be higher due to strain hardening. Applicability of Volume Constancy:The relationship σₜ = σ(1 + ε) is derived assuming volume constancy, which may not be strictly true for all materials or deformation processes. Some materials experience significant changes in volume during plastic deformation.
How do you calculate hoop stress in thin-walled cylinders?
Hoop stress (σh), also known as circumferential stress, in a thin-walled cylinder subjected to internal pressurepis calculated using the following formula:
σh = (p r) / t
Where:
p is the internal pressure.
r is the radius of the cylinder.
t is the wall thickness.
This formula provides thenominalhoop stress. For thick-walled cylinders, more complex formulas, such as the Lamé equations, are necessary. True stress considerations are important if the cylinder experiences large plastic deformations.
What is the difference between true stress and engineering stress?
As explained above, engineering stress, or nominal stress, is calculated using the original cross-sectional area, while true stress is calculated using the instantaneous cross-sectional area. Engineering stress is simpler to calculate but becomes inaccurate at large strains, while true stress provides a more realistic representation of the stress state within the material during deformation.
When should principal stress formulas be applied in design?
Principal stresses (σ₁ and σ₂) represent the maximum and minimum normal stresses at a point, acting on planes where the shear stress is zero. Principal stress formulas are applied in design when dealing with multi-axial stress states, such as those found in complex geometries or under combined loading conditions. Knowing the principal stresses is crucial for predicting failure, especially when using failure theories like the von Mises or Tresca criteria, which are based on the stress differences related to the principal stresses. They are also essential for determining the maximum shear stress, which can be important for ductile materials. It's important to note that these formulas are typically used withnominalstresses; however, when large deformations are present, consideration should be given to using true stress.
Conclusion
The choice between using nominal stress and true stress depends on the specific application and the extent of plastic deformation involved. Nominal stress offers a simplified approach suitable for small deformations and preliminary estimates, while true stress provides a more accurate representation of the stress state under large plastic deformations. Understanding the nuances of both concepts and their respective limitations is crucial for engineers and materials scientists to ensure the safety and reliability of structures and components. By carefully considering the material behavior and the loading conditions, engineers can make informed decisions regarding stress calculations and design strategies.