In structural analysis, particularly when components endure high loads exceeding their yield strength, accurate stress-strain curves are indispensable for precise calculations. However, obtaining these curves can pose challenges, **often requiring approximations in the absence of readily available data**. This blog post delves into one method of stress-strain curve approximation that closely mirrors reality.

### Exploring Curve Simplifications

Before delving into the preferred method, let’s explore existing simplifications commonly used when precise data is unavailable. These include the **Perfectly Plastic Rigid Curve, Perfectly Plastic Elastic Curve, and Elastic with Linear Strain Hardening Curve**.

**Perfectly Plastic Rigid**: This model assumes a rigid tensile specimen that experiences no elastic deformation until it reaches its yield strength. Once the yield strength is attained, the material undergoes plastic strain at a constant stress level, without any strain hardening effect.

**Perfectly Plastic Elastic**: In this simplification, stress and strain maintain a linear relationship within the elastic region until the material reaches its yield point. Beyond this threshold, immediate plastic deformation occurs at a constant stress level, without any strain hardening.

**Elastic with Linear Strain Hardening**: This model depicts a material exhibiting initial elastic behavior, characterized by a linear stress-strain relationship. Upon surpassing the yield point, plastic deformation commences, accompanied by strain hardening, which also follow a linear stress-strain relationship.

### Introducing the Ramberg–Osgood Relationship

The Ramberg–Osgood equation was created to **describe the nonlinear relationship between stress and strain** — that is, the stress–strain curve — in materials **after their yield points**. It is especially applicable to **ductile metals that harden with plastic deformation**, showing a smooth elastic-plastic transition.

The equation is expressed as:

Here, ε represents total strain, σ signifies stress, E stands for Young’s Modulus, while K and n are material-dependent constants.

In the above expression, the term ** σ/E is the elastic strain** portion and the term

**portion.**

`K(σ/E)`^{n}

is the plastic strain It’s crucial to note that this equation relates** engineering stress with engineering strain**, necessitating a transformation to **true stress and true strain** before utilization in Finite Element Analysis (FEA) software.

### The H. N. Hill Equation

While the Ramberg–Osgood relationship offers precision, **determining the parameters K and n can be challenging**. To address this, H. N. Hill proposed an adaptation of the equation, simplifying parameter estimation by incorporating yield strength directly.

The H. N. Hill Equation is expressed as:

Where σ_{YS} denotes yield strength.

Notice in the expression above that the elastic strain component, represented by σ/E, remains unchanged. However, the plastic strain component, given by 0.002(σ/σ_{YS})^{n}, has undergone alteration, resulting in the absence of the term “K.” Instead, the yield strength is now present, which is a readily accessible material property.

Let’s revisit the definition of yield strength. Yield strength denotes the maximum stress a material can sustain without experiencing **considerable** permanent deformation, typically characterized by a **plastic strain of 0.2%**. This parameter is crucial in indicating the threshold at which a material shifts from elastic to plastic behavior under tensile or compressive loading. Once the yield strength is surpassed, the material undergoes plastic deformation (even though there’s already 0.2% plastic strain, we usually set it to zero), indicating that it will not revert to its original shape after the removal of the load. It’s important to emphasize that the **proportional limit** is actually the stress the ends the elastic region and that presents zero plastic strain.

In the H. N. Hill Equation, this definition is considered: **when stress equals the yield strength, the corresponding plastic strain is 0.2%**. Consequently, **the stress at which the plastic strain is zero** **will be lower than the yield strength**. This stress is known as the **proportional limit**.

### Calculating the Parameter “n”

Determining the parameter “n” is crucial for curve accuracy. Utilizing known mechanical properties like **Ultimate Tensile Strength (σ _{UTS}) and Maximum Elongation (ε_{max})**, “n” can be calculated using the equation:

Isolating “n”:

### Converting to True Stress-Strain

As engineering stress-strain curves are commonly used, conversion to true stress-strain curves is essential for FEA software. The conversion is achieved through the equations:

## Example with ASTM A36

Let’s delve into a step-by-step process using ASTM A36 as an example, one of the most common materials. The following mechanical properties can be easily found on the internet:

**Step 1: Calculate “n”**

To begin, we calculate “n” using known mechanical properties:

**Step 2: Derive the Engineering Stress/Strain Curve**

Utilizing the derived “n,” we obtain the engineering stress-strain curve:

**Step 3: Conversion to True Stress/Strain**

Converting engineering values to true stress/strain involves:

For example, with engineering stress of 300 MPa and strain of 0.01338960 from the table:

We can compute these values for all data points in the table above. By superimposing these two charts, we can visually discern the disparity between engineering and true stress/strain. The resulting comparison is illustrated below:

Notice that at small strains, the two curves closely resemble each other. However, **as strain increases, the disparity between them becomes significant.** This occurs because **true stress calculation takes into account the instantaneous cross-sectional area**, whereas **engineering strain only considers the initial cross-sectional area**. When a body undergoes deformation from a tensile load, the cross-sectional area decreases due to the Poisson’s effect. As stress is defined as force divided by area, a decrease in area results in higher stress. Consequently, true stress exceeds engineering stress for large strains.

The new table generated for true stress/strain is:

Interested in delving deeper into engineering and true stress/strain? Explore this insightful blog post on the topic by clicking here.

**Step 4: Extraction of True Stress vs. True Plastic Strain Curve**

We currently possess the True Stress vs Total Strain Curve. In a finite element method software, the appropriate curve to employ is the **true stress versus true plastic strain curve.** Consequently, we must exclude the true elastic strain component from the prior curve.

Let’s employ the expression provided, starting from the line on the table with a stress of 250.8 MPa (True Yield Strength) and a strain of 0.00324473, and continue to the final line, where the stress reaches 480 MPa and the strain is 0.18232156. The calculated result is presented below:

**Step 5: Calculate True Stress for Zero True Plastic Strain**

To ensure compatibility with FEA software, **the first line of the table should feature a true stress value corresponding to zero true plastic strain**. This value can be determined through interpolation using the first two points of the preceding table.

Thus, we obtain a revised table suitable for FEA software integration:

Acknowledged, it’s important to note that **the stress value for zero plastic strain does not align with the yield strength**. Additionally, at a true plastic strain of 0.001995511 = 0.01995511% (second line of the table above), the stress equals 250.8 MPa, which matches the **true yield strength** (while the engineering yield strength remains at 250 MPa). **The discrepancy in the plastic strain for yield strength arises because the strain was converted to true strain**, resulting in a slightly different value from the nominal 0.2%.

**Important Note**

By employing this method to derive the true stress/strain curve, **it’s feasible to incorporate up to 0.2% plastic strain into your model while still assuming that the part has not yielded**. This assertion holds true because only at a true plastic strain of 0.001995511 = 0.1995511% (rounded to 0.2%) do we observe a stress value equal to the material’s yield strength.

**Explore Further with Python**

We’ve developed a Python script to handle these calculations for you, offering results with enhanced accuracy thanks to additional features. Here’s what sets it apart:

- Optimal stress distribution is considered up to the ultimate tensile strength.

- A more precise point for linear interpolation in step 5 is taken into account.

- Beyond the ultimate tensile strength, the script establishes a
**linear function with a slope equal to 10% of the slope of the last two points**. This conservative approach ensures safer calculations at higher stress levels. Without this adjustment, the software would simply rely on the slope of the last two points.

Feel free to download the Python code by clicking here.

## Learn More

Enhance your understanding of FEA with the following blog posts:

## References

- Ramberg W and Osgood W. R. – Description of Stress-Strain Curves by Three Parameters – No. 9023, NACA, 1943

- Hill H. N. – Determination of Stress-Strain Relations from “Offset” Yield Strength Values

- Office of Aviation Research – DOT/FAA/AR-MMPDS-01 – Metallic Materials Properties Development and Standardization

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