Alexei Volkov
Plastic strain energy density is a critical parameter in understanding the mechanical properties of materials, particularly in the design and analysis of mechanical systems. It is a measure of the energy absorbed by a material during deformation, specifically the energy per unit volume that a material can absorb without undergoing permanent deformation. The calculation of plastic strain energy density is essential in assessing the fatigue strength and life of structures, especially under cyclic and random loading conditions. Various methods and models have been proposed to calculate plastic strain energy density, including the Ramberg-Osgood equation, the instantaneous power history approach, and the use of stress-strain curves. These calculations are crucial in predicting and improving the performance of materials in applications such as ship structures and alloys.
| Characteristics | Values | |
|---|---|---|
| Calculation of plastic strain energy density | Requires the calculation of the sum of hysteresis loops for nine components of the stress and strain state | |
| Calculation of ΔWe+ | Requires observation of changes in stress history σij(t) and integration of the relationship ε=σ(ε) only for positive values of σij(t) | |
| Plastic strain | Permanent deformation that remains after the removal of stresses; generally time- and rate-dependent | |
| Plastic skeleton-strain | The larger of the absolute values of the plastic strain components of the skeleton curves in the tension and compression stress sides | |
| Accumulated plastic strain | The sum of plastic tension and compression strains in a cyclic stress-strain curve | |
| Plastic strain rate | $\dot{\epsilon_}^Equivalent plastic strain rate</co: 16> | \(\dot{\bar{\epsilon}}=\sqrt{\frac{2}{3}\dot{\epsilon_{ij}}^{p}\dot{\epsilon_{ij}}^{p} }\) |
| Elastic unloading | Releasing the stress on a body that is at its yield point | |
| Neutral loading | The loading is tangent to the yield surface | |
| Plastic loading | The body is loaded plastically | |
| Plastic skeleton-strain and accumulated plastic strain | Can be used to evaluate the reduction in absorbed energy | |
| Plastic strain energy density | Can be calculated by integrating Equation (3) with respect to strain, as shown in Equation (4) | |
| Single-cycle plastic strain energy density | \(\frac{1}{2}\left[\sigma_{f}\left(\epsilon_{p a}^{2}-\epsilon_{f}^{2}\right)+\sigma_{0,2}\left(\epsilon_{0,2}-\epsilon_{f}\right)^{2}\right]\) |
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What You'll Learn
Calculating plastic strain energy density under cyclic and random loading
The calculation of plastic strain energy density under cyclic and random loading is a complex topic that has been the subject of numerous studies. The primary objective of these studies is often to assess the low cycle fatigue (LCF) behaviour of alloys and understand the dynamic structural changes during cyclic deformation.
One approach to calculating plastic strain energy density under cyclic and random loading is to use the strain energy density approach (SED). This method involves observing stress changes and integrating suitable fragments of the instantaneous power history according to the distinguished ranges of loading changes. It can be applied to both cyclic and random loading without any limitation for loading causing elastic and elastic-plastic strains.
Another method is to use the Ramberg-Osgood equation to approximate the stress-strain curve and then integrate the area under the curve. This calculation provides a better estimate of the modulus of toughness, which is the amount of strain energy per unit volume that a material can absorb without permanent deformation. The units of the modulus of toughness are the same as the strain energy density, which are psi in US Customary units and Pa in SI units.
Additionally, a statistical method has been proposed for estimating plastic strain energy per cycle in metals experiencing fatigue. This method incorporates a two-parameter Weibull probability function, where monotonic stress-strain data is used as input. The results obtained through this method have been verified using a thermodynamically-based model.
Furthermore, when dealing with random loading, it is important to calculate the sum of hysteresis loops for the nine components of the stress and strain state. This calculation can be challenging due to the need to interpret the strain energy corresponding to open hysteresis loops.
Overall, the calculation of plastic strain energy density under cyclic and random loading involves a combination of experimental data, mathematical models, and observations of stress-strain relationships to understand and predict the behaviour of materials under different loading conditions.
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The Ramberg-Osgood equation
The equation is expressed as:
${\displaystyle \varepsilon ={\frac {\sigma }{E}}+\alpha {\frac {\sigma }{E}}\left({\frac {\sigma }{\sigma _{0}}}\right)^{n-1}}$
Or:
${\displaystyle \varepsilon = {\frac {\sigma }{E}} + \left({\frac {\sigma }{K}} \right)^n}$
Where:
- ${\displaystyle \varepsilon}$ is the total strain
- ${\displaystyle \sigma}$ is the stress
- ${\displaystyle E}$ is the elastic modulus
- ${\displaystyle \alpha}$ and ${\displaystyle K}$ are material-dependent parameters describing the hardening behaviour of the material
- ${\displaystyle n}$ is the inverse of Hollomon's strain hardening coefficient
While the Ramberg-Osgood equation is a useful approximation, it's important to note that it doesn't perfectly represent the real stress-strain behaviour of materials. This is because it implies that plastic strain is present at any stress level, even below the yield point. However, the plastic strain component of the total strain is typically negligible at low-stress levels.
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Plastic strain energy density in low-cycle fatigue tests
The plastic strain energy density in low-cycle fatigue tests is a critical parameter in understanding the fatigue behaviour of materials. It provides insights into the accumulation and dissipation of energy during cyclic loading, which is essential for predicting the fatigue life of components subjected to repeated loading and unloading.
One of the earliest methods to calculate plastic strain energy was proposed by Esin and Jones, who utilised the monotonic tensile stress-strain curve to estimate fatigue life in low-cycle and high-cycle regimes. They employed an electric circuit to identify the onset of microplasticity, where changes in resistance as a function of strain become non-linear. However, this procedure was time-consuming and challenging to implement.
To address these limitations, Korsunsky et al. introduced a microscopic energy dissipation criteria (EDC) approach for calculating the plastic strain energy per cycle. This method involved modelling the micro-scale elastoplastic behaviour through crystal slip modelling, revealing that the microscopic energy dissipation density was location-dependent.
The plastic strain energy density under uni-axial tension and cyclic loading can be derived using the modified Ramberg-Osgood constitutive relation, which characterises the stress-strain relationship of strain-hardening materials. This relation has been successfully applied to aluminium alloys LC9 and LY12, demonstrating agreement with experimental results.
Furthermore, the strain energy density approach (SED) is a valuable tool for assessing the fatigue strength of circumferential V-notched specimens under axial loading and rotating bending. The SED method can be extended to include elasto-plastic considerations, reducing the scatter index of the evaluated equivalent strain energy density. This energy-based approach offers a practical engineering tool for fatigue assessment.
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Plastic strain rate
The plastic strain rate is a fundamental concept in the study of mechanical properties of materials. It plays a crucial role in understanding the behaviour of materials under various loading conditions. The plastic strain rate is influenced by factors such as stress, temperature, and phase change. It is important to note that the plastic strain rate is different from the elastic strain rate, which is associated with the elastic response of a material.
The plastic strain rate can be determined using different methods, depending on the specific material and loading conditions. One common approach is to utilise the Cowper-Symonds equation, which accounts for the non-linear relationship between stress and strain. The equation is particularly useful in simulating ship collision scenarios, where the plastic strain rate is a critical factor influencing the structural integrity of ships. The material constants within the Cowper-Symonds equation, denoted as "D" and "q", vary depending on the type of material being analysed. For instance, in the context of mild steels, the recommended values are D = 40.4 s^-1 and q = 5, whereas for high-tensile steels, the values differ, with D = 3200 s^-1 and q = 5.
It is worth noting that in rate-dependent plasticity, the plastic response of a material is contingent upon the loading rate. This is in contrast to rate-independent plasticity, where the yield condition remains constant regardless of loading rates. When dealing with rate-dependent materials, an overstress model can be employed to describe the plastic response, taking into account the influence of stress, temperature, and phase changes.
Additionally, the plastic strain rate can be calculated using the Kuhn-Tucker conditions, particularly in the context of rate-independent plasticity with isotropic hardening. This method involves a set of algebraic equations that relate stress, strain, and the material's properties. Furthermore, the plastic strain rate is also influenced by the ductility of the material, which is inversely related to its strength and stiffness. As the strain rate increases, the ductility tends to decrease, impacting the overall behaviour of the material.
In summary, the plastic strain rate is a critical parameter in understanding the mechanical behaviour of materials, particularly in the context of plasticity and deformation. Its calculation involves considering various factors, including stress, temperature, phase changes, and the inherent properties of the material. By utilising equations such as the Cowper-Symonds equation and applying principles of rate-dependent plasticity, engineers can analyse and predict the response of materials under different loading conditions.
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Plastic skeleton-strain and absorbed energy
Plastic deformation is a type of deformation that results in permanent alterations in the size and shape of a specimen due to changes in the microstructure of a material. Materials that undergo plastic deformation are unable to return to their original size and shape after the load is removed. This is in contrast to elastic deformation, where materials can recover their initial size and shape.
Plastic skeleton-strain refers to the ability of a material to resist deformation, known as the modulus of resilience. It is graphically represented by the slope of a stress-strain curve, with stress on the y-axis and strain on the x-axis. The area under the stress-strain curve at any level of deformation represents the strain energy density, which is the strain energy per unit volume stored in the material. This energy is absorbed by the material, leading to rearrangements in the atoms within the material.
The calculation of plastic strain energy density can be complex, especially under cyclic and random loading conditions. The strain energy density approach (SED) is a method used to assess the energy absorption capacity of materials under these conditions. It involves observing stress changes and integrating suitable fragments of the instantaneous power history. The Ramberg-Osgood equation can also be used to approximate the stress-strain curve and calculate the plastic strain energy density by integrating the area under the curve.
The ability of a material to absorb and store strain energy without breaking or permanently deforming is an important property. This property allows materials to transfer energy from one part to another or to store and release energy. The energy absorbed during plastic deformation is consumed and expelled from the system as the microstructure of the material is irreversibly changed. This results in a reduction in the energy available for mechanical work, as some energy is transferred to the environment as heat.
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Frequently asked questions
The formula for calculating plastic strain energy density is ε = σ/E0 + εf.
The plastic strain energy density formula for cyclic loading is ε0.2 = σ0.2/E0.2 + 0.002.
The plastic strain rate is calculated using the formula $\dot{\epsilon_{ij}}^{p}$.
The formula for the equivalent plastic strain rate is $\dot{\bar{\epsilon}}=\sqrt{\frac{2}{3}\dot{\epsilon_{ij}}^{p}\dot{\epsilon_{ij}}^{p}}$.
The Ramberg-Osgood equation is a good method for calculating plastic strain energy density. This method involves approximating the stress-strain curve and then integrating the area under the curve.
