Alexei Volkov
Plastic strain is the non-recoverable strain generated when the loading exceeds the stress beyond the proportional limit in a stress-strain curve. It is important to understand a material's plastic strain ratio and how to measure it to accurately establish its formability. The plastic strain ratio is calculated using the ASTM E517 Standard Test Method, which depends on the ratio of width and thickness changes as the sample is pulled in tension. The plastic strain at failure can be calculated from the percent elongation. The critical equivalent plastic strain is the maximum strain that can be borne to ensure perforation stability. Accumulated plastic strain is the general criterion used to determine the amount of plastic loading on materials, which must be maintained within certain limits to avoid unstable fracture or collapse.
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What You'll Learn
Calculating plastic strain values for sheet metals
The plastic strain ratio, also known as the r-value, is a crucial parameter for sheet metals as it indicates their ability to resist thinning or thickening when subjected to tensile or compressive forces. This resistance to deformation is essential for various applications, especially in the automotive and aerospace industries, where sheet metals are formed into complex shapes.
To calculate the plastic strain ratio for sheet metals, ASTM E517-00 provides a standardised test procedure. This method involves measuring the thickness and breadth uniformity of each specimen in the gauge section. The r-value is then determined based on the ratio of width and thickness changes as the sample is pulled in tension. It's important to note that the volume of the specimen is assumed to remain constant, and the thickness strain can be expressed as et = ln(Lowo/Lfwf).
There are two methods for calculating the plastic strain ratio: the automatic method and the manual method. The automatic method involves pulling the specimen until it fractures, allowing for the determination of ultimate strength, yield strength, and elongation in a single test. To calculate plastic strains, you subtract the elastic strains from the measured strains. However, this method is sensitive to errors in width measurement, and edge curling can impact the accuracy of results.
The manual method, on the other hand, involves using calipers or a single extensometer to measure the plastic strain ratio. While this method may be more time-consuming, it can provide more accurate results if performed carefully.
Additionally, it's important to compensate for specimen curling and use rounded or flat knife edges to eliminate errors in the r-value. The orientation of the test specimens is also significant, and samples should be cut at 0 degrees, 45 degrees, and 90 degrees relative to the rolling direction to account for planar anisotropy in rolled sheet metals.
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The Ramberg-Osgood equation and plastic strain
The Ramberg-Osgood equation describes the nonlinear relationship between the stress and strain of a material around its yield point. The total strain is the sum of the elastic strain and the plastic strain. The equation is expressed as:
$$ {\displaystyle \varepsilon ={\frac {\sigma }{E}}+\alpha {\frac {\sigma }{E}}\left({\frac {\sigma }{\sigma _{0}}}\right)^{n-1}} $$
Where:
- $ {\displaystyle \varepsilon }$ is the total strain
- $ {\displaystyle {\frac {\sigma }{E}}}$ is the elastic part of the strain
- $\alpha$ describes the hardening behaviour of the material
- ${\displaystyle n}$ is the inverse of Hollomon's strain-hardening coefficient
The Ramberg-Osgood equation provides a formula for obtaining strain, but determining the parameters K and n can be challenging. The H. N. Hill Equation is an adaptation of the Ramberg-Osgood equation that simplifies parameter estimation by incorporating yield strength directly.
The plastic strain ratio is an important factor in understanding a material's formability and can be determined through ASTM E517 Standard Test Method for metals. The ratio is calculated by measuring the changes in width and thickness as the sample is pulled in tension.
Equivalent plastic strain rate can be determined using the following equation:
$$ \dot{\bar{\epsilon}}=\sqrt{\frac{2}{3}\dot{\epsilon_{ij}}^{p}\dot{\epsilon_{ij}}^{p} } $$
Where:
- $ \dot{\bar{\epsilon}}$ is the equivalent plastic strain rate
- $ \dot{\epsilon_{ij}}^{p}$ is the plastic strain rate
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Plastic strain and the stress-strain curve
Plastic strain is a key consideration in understanding a material's mechanical properties and behaviour under stress. It is particularly important in designing reproducible forming operations, as it determines a material's formability.
The stress-strain curve is a graphical representation of the relationship between stress and strain in a material. It provides valuable insights into the material's behaviour during deformation and fracture. The curve is characterised by several distinct stages, each indicating different properties.
The first stage is the linear elastic region, where the stress is directly proportional to the strain, following Hooke's law. In this region, the material undergoes elastic deformation, and the slope of the curve is known as Young's modulus. At the end of this stage, the material reaches the yield point, marking the initiation of plastic deformation.
The second stage is the strain hardening region, where the stress surpasses the yielding point and continues to increase until it reaches the ultimate tensile strength (UTS). This is the maximum stress that the material can withstand before fracture. During this stage, the material's cross-sectional area decreases uniformly, and necking may occur, leading to a "cup and cone" fracture characteristic of ductile materials.
The area under the stress-strain curve represents the energy absorbed during deformation and fracture, known as resilience in the elastic regime and toughness in the plastic regime. This area helps classify materials as brittle or ductile.
The plastic strain ratio is a critical parameter in understanding a material's behaviour. It is calculated using ASTM E517 Standard Test Method for plastic sheet metals. The ratio depends on the width and thickness changes as the sample is subjected to tension. By assuming a constant volume, the thickness strain can be expressed as et = ln(Lowo/Lfwf). Substituting this into the equation allows for the calculation of the plastic strain ratio.
Additionally, accumulated plastic strain is an important consideration. It refers to the total plastic deformation a material experiences and must be maintained within certain limits to prevent unstable fracture or collapse. An allowable accumulated plastic strain level of 2% is generally recommended, but recent tests on modern pipeline steel indicate that up to 5% plastic strain can be acceptable.
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Plastic strain and the elastic modulus
Plastic strain is a measure of how much a metal can deform before it thins or fractures. It is a crucial factor in understanding a material's formability and designing a reproducible forming operation. The plastic strain ratio is calculated using the ASTM E517 Standard Test Method, which takes into account the ratio of width and thickness changes as a sample is pulled in tension.
Elastic modulus, on the other hand, is a measure of a material's elasticity, or its ability to return to its original shape after deformation. It is one of the two parameters that determine a material's elasticity, the other being its elastic limit. A high elastic modulus indicates that a material is hard to deform, while a low elastic modulus indicates that a material is easily deformed. For example, a steel band typically has a high elastic modulus, whereas a rubber band has a low elastic modulus.
The elastic modulus is also used in the Ramberg-Osgood equation, which approximates the stress-strain curve for a material. This equation takes into account the yield strength, ultimate strength, elastic modulus, and percent elongation of the material.
When stress is applied to a material, it can undergo elastic or plastic deformation. Elastic deformation occurs when the stress is within the elastic limit, and the material returns to its original shape and size when the load is removed. Plastic deformation occurs when the stress exceeds the elastic limit, and the material becomes permanently deformed and does not return to its original shape and size.
The distinction between elastic and plastic deformation is important in understanding the behaviour of materials under load. It is also crucial in determining the suitability of materials for specific applications, such as in the design of steel tubes for umbilical systems, where accumulated plastic strain must be maintained within certain limits to avoid unstable fracture.
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Plastic strain and the plastic strain ratio
Plastic strain is the deformation a material undergoes that exceeds its elastic limit. This deformation is permanent and is also known as plastic deformation. The plastic strain ratio is a measure of a material's formability and is used to evaluate the drawability of sheet metals.
The plastic strain ratio is calculated using the ASTM E517 Standard Test Method for Plastic Strain Ratio r for Sheet Metal. The r-value is dependent on the ratio of width and thickness changes as the sample is pulled in tension. The volume of the specimen is assumed to remain constant, and the thickness strain is expressed as et = ln(Lowo/Lfwf). After substituting et into Equation 1 and inverting it to eliminate negative values, the plastic strain ratio is given by Equation 2.
The plastic strain ratio can be calculated manually with a set of calipers or automatically with two extensometers. The automatic method involves pulling the specimen until it fractures, which enables the determination of ultimate strength, yield strength, and elongation in a single test. However, errors in the r-value can occur due to edge curling and concentrated stresses, resulting in higher width strains and higher r-values.
The r-value remains constant over a range of plastic strains for many materials, but it varies with the applied axial strain for some sheet materials. Rolled sheet metals exhibit planar anisotropy, which means that sample orientation can significantly impact the measurement of the plastic strain ratio. Therefore, it is crucial to cut test specimens at 0 degrees, 45 degrees, and 90 degrees relative to the rolling direction and report the cut direction with each result.
The plastic strain ratio is essential in designing reproducible forming operations. By testing the incoming sheet material, manufacturers can ensure consistent part quality and scrap rate. Additionally, accumulated plastic strain is a critical factor in determining the effect of reeling and cyclic bending on pipelines and umbilical steel tubes. Maintaining acceptable levels of accumulated plastic strain prevents unstable fractures or plastic collapse in steel tubes during fabrication and installation.
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Frequently asked questions
Plastic strain is the permanent deformation in a material that has been loaded beyond its elastic limit.
Plastic strain can be calculated from total strain using the equation: σ = H εpn. The plastic strain at failure can be calculated from the percent elongation, eL, by εf = eL/100%.
The plastic strain ratio is the ratio of width and thickness changes as a sample is pulled in tension. It is calculated using ASTM E517 Standard Test Method for Plastic Strain Ratio r for Sheet Metal.
The plastic strain ratio can be calculated manually with a set of calipers or automatically with extensometers. The automatic method involves pulling the specimen until it fractures, which determines the ultimate strength, yield strength, and elongation in a single pull.
The equivalent plastic strain is the maximum equivalent strain that can be borne to ensure perforation stability. It is defined as: $\dot{\bar{\epsilon}}=\sqrt{\frac{2}{3}\dot{\epsilon_{ij}}^{p}\dot{\epsilon_{ij}}^{p}}$.
