Engineering Stress Formula vs. True Stress Formula: A Comprehensive Guide
In the field of mechanical engineering, understanding stress is paramount to ensuring the safety and reliability of structures and machines. Two fundamental concepts, engineering stress and true stress, are used to quantify the internal forces acting within a deformable body subjected to external loads. While both relate force to area, they differ significantly in how they account for the changing geometry of the material during deformation. Choosing the correct stress measure is crucial for accurate analysis and prediction of material behavior, especially under large deformation scenarios. This article delves into the definitions, applications, and differences between engineering stress and true stress, providing a comprehensive guide for engineering students, practicing engineers, and researchers.
Defining Engineering Stress
Engineering stress, often denoted as σ (sigma), is defined as the force applied to a material divided by its original cross-sectional areabeforeany deformation occurs. The formula for engineering stress is:
σ = F/A₀
where: σ = Engineering stress (typically in Pascals (Pa) or pounds per square inch (psi))
F = Applied force (typically in Newtons (N) or pounds (lb))
A₀ = Original cross-sectional area (typically in square meters (m²) or square inches (in²))
This definition is straightforward and easy to apply, making it a common choice for many engineering calculations. However, it's important to recognize its limitations, particularly when dealing with ductile materials undergoing significant plastic deformation.
Defining True Stress
True stress, often denoted as σt (sigma subscript t), is defined as the force applied to a material divided by itsinstantaneouscross-sectional areaduringdeformation. The formula for true stress is:
σt = F/A
where: σt = True stress (typically in Pascals (Pa) or pounds per square inch (psi))
F = Applied force (typically in Newtons (N) or pounds (lb))
A = Instantaneous cross-sectional area (typically in square meters (m²) or square inches (in²))
As the material deforms, its cross-sectional area changes, especially in the case of necking during tensile testing of ductile materials. True stress provides a more accurate representation of the stress experienced by the material at any given point during the deformation process.
Key Differences and When to Use Each
The core difference between engineering stress and true stress lies in the area used in the calculation. Engineering stress uses the original area, while true stress uses the instantaneous area. This difference becomes significant when the material undergoes large plastic deformation.
Engineering Stress: Simpler to calculate, based on readily available initial dimensions. Adequate for situations with small elastic deformations. Commonly used for design calculations where the allowable stress is well below the yield strength of the material, and the deformation is minimal. True Stress: More accurate representation of the actual stress state within the material, especially at large deformations. Essential for understanding material behavior during processes like metal forming, where significant shape changes occur. Required for accurate modeling of material behavior during failure.
During a tensile test, for example, the engineering stress will typically reach a maximum value (the tensile strength) and then decrease as necking occurs. This decrease is an artifact of the calculation; theactualstress within the necking region is still increasing. True stress, by accounting for the decreasing area, will continue to increase until the material fractures.
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 pressurep, is calculated using the following formula:
σh = (p r) / t
where: σh = Hoop stress
p = Internal pressure
r = Radius of the cylinder
t = Wall thickness of the cylinder
This formula assumes that the wall thickness is significantly smaller than the radius (typically t < r/10). It's important to note that this formula calculates theengineeringhoop stress. Because the cylinder walls deform slightly under pressure, atruehoop stress could also be calculated, although the difference is often negligible in thin-walled cylinders under normal operating conditions.
What is the difference between true stress and engineering stress?
As summarized above, the key difference lies in the area used for calculation. Engineering stress uses the original cross-sectional area (A₀), whereas true stress uses the instantaneous cross-sectional area (A) at the moment of measurement. This distinction is crucial during large plastic deformations.
When should principal stress formulas be applied in design?
Principal stress formulas are essential when designing components subjected to multi-axial stress states (i.e., stress acting in multiple directions simultaneously). These formulas determine the maximum and minimum normal stresses (principal stresses) and the maximum shear stress acting on the material, regardless of the orientation of the coordinate system.
Principal stresses are used to: Predict yielding: Yielding occurs when the maximum shear stress or a combination of principal stresses exceeds the material's yield strength, as defined by yield criteria like the von Mises or Tresca criteria. Analyze fatigue failure: Fatigue life is influenced by the range of principal stresses experienced during cyclic loading. Determine fracture initiation:Fracture often initiates at locations where the principal stresses are highest, particularly in brittle materials.
Principal stress formulas are particularly important in complex geometries and loading scenarios, such as pressure vessels with nozzles, rotating machinery with stress concentrations, and structural components with complex boundary conditions. The formulas involve finding the eigenvalues of the stress tensor, a mathematical representation of the stress state at a point.
Relationship Between Engineering Stress and True Stress
Under conditions of constant volume (plastic deformation often approximates constant volume), there is a direct relationship between engineering stress and true stress. This relationship is derived from the conservation of volume:
A₀L₀ = AL
where:
A₀ = Original cross-sectional area
L₀ = Original length
A = Instantaneous cross-sectional area
L = Instantaneous length
The engineering strain (ε) is defined as:
ε = (L - L₀) / L₀
And the true strain (εt) is defined as:
εt = ln(L / L₀)
From the volume conservation equation, we can derive:
A/A₀ = L₀/L = 1 / (1 + ε)
Therefore, the relationship between true stress and engineering stress becomes:
σt = σ (1 + ε)
This equation is valid until necking occurs. After necking, the deformation becomes highly localized, and the assumption of uniform deformation is no longer valid.
Examples and Applications
Example 1: Tensile Testing of a Copper Wire
A copper wire with an initial diameter of 2 mm and a length of 100 mm is subjected to a tensile force. At a certain point, the force is 500 N, and the diameter of the wire is measured to be 1.8 mm. Calculate the engineering stress and the true stress.
Step 1: Calculate the original area (A₀).
A₀ = π (d₀/2)² = π (2 mm / 2)² = π (1 mm)² = 3.1416 mm² =
3.1416 x 10⁻⁶ m² Step 2:Calculate the engineering stress (σ).
σ = F/A₀ = 500 N / (3.1416 x 10⁻⁶ m²) =
159.15 MPa Step 3:Calculate the instantaneous area (A).
A = π (d/2)² = π (1.8 mm / 2)² = π (0.9 mm)² =
2.5447 mm² =
2.5447 x 10⁻⁶ m² Step 4:Calculate the true stress (σt).
σt = F/A = 500 N / (2.5447 x 10⁻⁶ m²) =
196.46 MPa
The true stress (196.46 MPa) is significantly higher than the engineering stress (159.15 MPa) due to the reduction in the cross-sectional area.
Example 2: Pressure Vessel Design
Consider a cylindrical pressure vessel made of steel with an internal diameter of 1 meter and a wall thickness of 10 mm. The vessel is designed to withstand an internal pressure of 5 MPa. Calculate the hoop stress using both engineering stress and a modified true stress approach. For simplification assume the change in radius is negligible and only consider the change in thickness. The hoop strain is approximated as 0.025. Step 1:Calculate the engineering hoop stress.
σh = (p r) / t = (5 MPa 0.5 m) /
0.01 m = 250 MPa Step 2: Calculate the approximate instantaneous thickness. Since ε = (t-t₀)/t₀ then t = t₀ + ε t₀ = t₀ (1+ε) =
0.01m (1 +
0.025) =
0.01025m. Step 3:Calculate the true hoop stress using the adjusted thickness.
σht = (p r) / t = (5 MPa 0.5 m) /
0.01025 m =
243.90 MPa
In this case, the difference between the engineering and approximate true stresses is relatively small (approximately 2.4%), suggesting that for small deformations, the engineering stress provides a reasonably accurate estimate. However, in situations with higher pressures or thinner walls, the difference would become more significant, warranting the use of the true stress formula for a more accurate assessment.
Common Pitfalls and Misconceptions
Assuming Engineering Stress is Always Sufficient: A common mistake is to rely solely on engineering stress calculations, particularly when dealing with ductile materials undergoing significant plastic deformation. This can lead to underestimation of the actual stress experienced by the material and potentially unsafe designs. Neglecting the Limitations of the Constant Volume Assumption: The relationship between engineering stress and true stress (σt = σ (1 + ε)) relies on the assumption of constant volume. While this is a good approximation for many plastic deformation processes, it may not be valid in all cases, especially when significant void formation or changes in density occur. Confusing True Stress with Principal Stress: While both involve the concept of 'true' values, they represent fundamentally different quantities. True stress is a measure of the stress accounting for the changing area, while principal stresses are the maximum and minimum normal stresses acting at a point, regardless of the orientation of the coordinate system. Applying Thin-Walled Cylinder Formulas to Thick-Walled Cylinders: The simple hoop stress formula for thin-walled cylinders (σh = (p r) / t) is not accurate for thick-walled cylinders (typically t > r/10). For thick-walled cylinders, more complex equations that account for the radial stress distribution through the wall thickness are required.
Conclusion
Understanding the difference between engineering stress and true stress is crucial for accurate stress analysis and reliable design in mechanical engineering. While engineering stress offers a simplified approach suitable for small deformations, true stress provides a more accurate representation of the stress state during large plastic deformations. Selecting the appropriate stress measure, considering the limitations of each, and applying relevant formulas are essential steps in ensuring the safety and performance of engineering structures and machines. By carefully considering these factors, engineers can make informed decisions and avoid potential pitfalls in their design and analysis work.