Alexei Volkov
Plastic strain is a permanent deformation that remains after the removal of stress. It is a critical factor in determining the formability of materials, especially in construction and engineering applications. The plastic strain ratio, or the r value, is a measure of how much a material can deform before it thins or fractures. This value is calculated by measuring the true strain in the width direction and the thickness direction when a material is pulled beyond its elastic limit. Various methods, such as manual calculations and automatic extensometers, can be employed to determine the plastic strain ratio. Additionally, factors like the yield point, stress-strain curves, and the type of loading (elastic, neutral, or plastic) come into play when understanding plastic strain. Understanding plastic strain-induced phase transformations, as observed in silicon, provides insights into the fundamental behaviour of materials under different types of compression.
| Characteristics | Values |
|---|---|
| Plastic strain | Permanent deformation |
| Plastic deformation | Material does not return to its original shape when unloaded |
| Plastic strain rate | \(\dot{\epsilon_{ij}}^{p}\) |
| Equivalent plastic strain rate | \(\dot{\bar{\epsilon}}\) |
| Plastic Loading | Loading the body plastically |
| Plastic strain-induced PTs | Occur at new defects constantly generated during plastic flow |
| Plastic strain ratio | r |
| Plastic strain ratio calculation | ASTM E517 Standard Test Method for Plastic Strain Ratio r for Sheet Metal |
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What You'll Learn
- Plastic strain is permanent and remains after stress removal
- Plastic strain increases yield stress and the yield/ultimate stress ratio
- Plastic strain-induced phase transformation in silicon
- Plastic strain ratio is a direct measure of sheet metal's drawability
- Plastic strain rate can be determined using the Kuhn-Tucker conditions
Plastic strain is permanent and remains after stress removal
Plastic strain is a critical concept in engineering and material science, referring to the permanent deformation of a material beyond its elastic limit. This deformation results in irreversible changes in the material's shape and size, even after the removal of applied stress. Understanding plastic strain is essential, as it influences how materials respond to applied loads and affects the structural integrity and performance of various materials under stress.
When a material undergoes plastic deformation, it does not return to its original form. This behaviour is in contrast to elastic deformation, where the material recovers its initial shape once the load is removed. In the context of a metal rod, for instance, slight bending may result in elastic deformation, allowing the rod to return to its original shape. However, if bent significantly, the rod undergoes plastic deformation and remains bent permanently.
The occurrence of plastic strain is associated with the yield point of a material. Every material possesses a yield point, which represents the stress level beyond which permanent deformation begins. Once the yield point is surpassed, any deformation that takes place is categorised as plastic strain. This phenomenon is a result of irreversible molecular changes within the material, such as alterations in atomic alignment and dislocation movements. These changes stabilise the material in its new form, preventing it from reverting to its original state.
The distinction between plastic and elastic strain is crucial in construction and engineering applications. For instance, in construction, materials that exhibit plastic deformation and fail to return to their original shape after unloading are usually unacceptable. A notable example is concrete, which is a brittle material. Its strength is defined by the stress at fracture rather than residual strain, making it unsuitable for applications that involve significant bending or deformation.
To summarise, plastic strain is permanent and persists even after stress removal. This characteristic sets it apart from elastic strain and has important implications in material science and engineering. By understanding plastic strain, engineers can make informed decisions about material selection and design, ensuring the structural integrity and performance of their creations.
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Plastic strain increases yield stress and the yield/ultimate stress ratio
Plastic strain is a permanent deformation that remains after the removal of stress. It is time- and rate-dependent. The accumulation of plastic strain is important in determining the effect of reeling, where cyclic bending plastic strain is considered for the multiple cycles within a reeling-unreeling cycle.
The yield point is the point at which a material exhibits plastic deformation and does not return to its original shape when unloaded. After the yield point, the material begins to strain harden, increasing its strength. This increase in strength is reflected in the stress-strain curve, where the strength of the material increases between the yield point and the ultimate strength.
The ratio of ultimate strength to yield strength, known as the strain hardening ratio, is used as a measure of the degree of strain hardening in a material. Accumulated plastic strain increases the yield stress of a material and, consequently, increases the yield/ultimate stress ratio. This is important to ensure safety, as a certain ratio between yield stress and ultimate tensile stress is desirable to prevent failure.
The yield stress of a material can vary depending on temperature and the presence of an impact or explosion. Additionally, ductile materials, such as some steels, may not have a clear yield point, making it challenging to determine the exact onset of plastic flow. However, the Rp0.2-criterion is used to address this issue, as it focuses on the gradual onset of yield due to dislocation movement in grains with favourable slip systems.
Overall, understanding the relationship between plastic strain and yield stress is crucial in engineering applications to ensure the safe use of materials and to optimize their performance under various conditions.
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Plastic strain-induced phase transformation in silicon
Firstly, plastic strain-induced phase transformations (PTs) occur at new defects generated during plastic flow, whereas pressure-induced PTs initiate at pre-existing defects like dislocations or grain boundaries. This distinction influences the required experimental characterization and thermodynamic and kinetic descriptions of the transformations. Plastic strain-induced PTs may occur at much lower pressures and follow strain-controlled kinetics, providing opportunities to explore hidden phases that are inaccessible under hydrostatic compression.
The mechanisms underlying plastic strain-induced PTs in silicon have been investigated, with a focus on the dislocation pileup-based mechanism (DPBM). This mechanism suggests that the concentration of stress components is directly proportional to the number of dislocations in a pileup, which can be substantial. The elaborated DPBM has been corroborated by experimental results, demonstrating its applicability in developing economic defect-induced synthesis of nanostructured materials and surface treatments.
Furthermore, the particle size of silicon plays a crucial role in the phase transformation pressure. The direct and inverse Hall-Petch effects describe the relationship between particle size, yield strength, and pressure for strain-induced PTs. For instance, in 100 nm particles, the strain-induced PT Si-I→Si-II initiates at 0.3 GPa under compression and shear, while under hydrostatic conditions, it requires 16.2 GPa. Additionally, the Si-I→Si-III PT commences at 0.6 GPa but does not occur under hydrostatic pressure.
The study of plastic strain-induced phase transformations in silicon has broad implications. It provides insights into the fundamental behaviour of silicon under various conditions and offers potential applications in fields such as photovoltaics, nanoelectromechanical systems, and solar energy conversion.
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Plastic strain ratio is a direct measure of sheet metal's drawability
The plastic strain ratio, or r-value, is a direct measure of sheet metal's drawability. It is a parameter that indicates a sheet metal's ability to resist thinning or thickening when subjected to tensile or compressive forces. This resistance contributes to the forming of shapes, such as cylindrical flat-bottom cups, by the deep-drawing process.
The r-value is the ratio of the true strain in the width direction to the true strain in the thickness direction when a sheet material is pulled in uniaxial tension beyond its elastic limit. The word "plastic" in the phrase "plastic strain ratio" implies that the specimen has exceeded its elastic limit and that only the strain that induces plastic flow is considered in the calculation.
The plastic strain ratio can be calculated manually with a set of calipers or automatically with two extensometers. The manual approach involves measuring the specimen's width and the distance between gauge marks before testing. The specimen is then pulled to a strain less than the maximum force, unloaded, and the final width and gauge length are measured. The automatic method involves pulling the specimen until it fractures, which allows for the determination of the ultimate strength, yield strength, and elongation in a single pull.
The r-value remains constant over a range of plastic strains up to the maximum force applied for many materials. However, for some sheet materials, the r-value varies with the applied axial strain. Therefore, the orientation of the specimen relative to the rolling direction is significant, and specimens should be cut at 0 degrees, 45 degrees, and 90 degrees respective to the rolling direction.
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Plastic strain rate can be determined using the Kuhn-Tucker conditions
The Kuhn-Tucker conditions are used to find the equivalent plastic strain rate, which is defined as:
$$ \dot{\bar{\epsilon}}=\sqrt{\frac{2}{3}\dot{\epsilon_{ij}}^{p}\dot{\epsilon_{ij}}^{p} } $$
Where, $ \dot{\bar{\epsilon}}$ is the equivalent plastic strain.
There are three cases to consider when determining the plastic strain rate:
- Elastic Unloading: This involves releasing the stress on a body at its yield point. In this case, $\dot{\bar{\epsilon}}^p = 0$, and $\dot{\mathbf{E}}:\mathbf{N}^p < 0$, resulting in $\dot{f} < 0$.
- Neutral Loading: This is when the loading is tangent to the yield surface, making the strain rate orthogonal to the direction of plastic flow. Here, $\dot{\mathbf{E}}:\mathbf{N}^p = 0$ and $\dot{\bar{\epsilon}}^p = 0$.
- Plastic Loading: This involves loading the body plastically, where the strain rate is no longer orthogonal to the direction of plastic flow.
The Kuhn-Tucker consistency condition at yield (which occurs at $f=0$) is an important consideration. This condition states that if $f = 0$, then $\dot{\bar{\epsilon}}^p \dot{f} = 0$.
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Frequently asked questions
Plastic strain is permanent and remains even after the removal of stresses. It is generally time and rate-dependent.
Understanding a material's plastic strain ratio and how to measure it are crucial in accurately establishing a material's formability. This is especially important for designing a reproducible forming operation.
The plastic strain ratio, r, is considered a direct measure of a sheet metal's drawability. It is calculated using Equation 1 or Equation 2, which involve measuring changes in width and thickness, as well as axial gauge length.
