Omar Khalid
The calculation of fully plastic stress is a complex process that involves understanding the behaviour of materials under various conditions. Most initially isotropic engineering materials exhibit elastic-plastic behaviour, which means that they can undergo irreversible deformation when subjected to loading. To calculate the stress in a plastic material, one can use the formula Stress = F / A, where F is the force and A is the area. This calculation can be further enhanced by considering variables such as temperature, which influences the flexibility and Young's modulus of plastics. Additionally, there are several methods available for estimating elastic-plastic stresses, including the Abdel-Karim-Ohno cyclic plasticity model and pseudocurve approaches. These methods offer valuable insights into the behaviour of materials under extreme conditions, such as impact loading, high pressures, and high temperatures.
| Characteristics | Values |
|---|---|
| Calculating elastic-plastic stress | The Abdel-Karim-Ohno cyclic plasticity model |
| Calculating plastic strain | Ramberg-Osgood expression using Hollomon parameters |
| Plasticity model | Multilinear or Bilinear |
| Multilinear modelling | Establishing properties as per the stress-strain curve obtained from a tensile test |
| Bilinear model | Does not take into account the entire behavioural history of the material |
| Bilinear model coefficients | Young modulus and plastic phase modulus |
| Plastic stress | Calculated by dividing force by cross-section: Stress = Force/Cross section |
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What You'll Learn
The Abdel-Karim-Ohno cyclic plasticity model
The Abdel-Karim-Ohno model has been shown to give acceptable predictions for all considered multiaxial conditions when used with an evolution function for μi. However, it gives poor predictions of uniaxial ratcheting if the parameter μi is determined from a multiaxial ratcheting response. The model has been implemented into commercial FE software, such as ANSYS, using a radial return method.
The Abdel-Karim-Ohno model is an advanced cyclic plasticity model called MAKOC, which also incorporates the isotropic hardening rule of Calloch and a memory surface introduced in a stress space in accordance with the Jiang-Sehitoglu concept. This model is useful for describing the behaviour of materials under non-proportional loading, which is not accurately captured by other cyclic plasticity models included in commercial FE software.
The MAKOC model has been compared with the Chaboche model included in some FE codes, and it has been found to provide better predictions for certain loading conditions. The model has been implemented into the FE code ANSYS 15.0 using Fortran subroutines for 1D, 2D, and 3D elements. This implementation allows for the study of steady-state material behaviour and its subsequent application in computational fatigue analysis.
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Multiaxial axial-torsion load paths
Multiaxial fatigue loading can be categorised into two types: proportional (in-phase) and nonproportional (out-of-phase). Proportionality is achieved when the peaks of multiaxial loads are reached at the same time, whereas nonproportionality refers to the peaks of multiaxial loads being out of sync. In the context of multiaxial axial-torsion load paths, this can involve the application of axial and torsional loads that are either in-phase or out-of-phase.
For instance, Jensen (2005) studied the multiaxial fatigue behaviour of a martensitic Nitinol alloy under force-controlled proportional tension-torsion tests. He conducted in-phase tension-torsion tests with combinations of fully reversed axial and shear stress amplitudes to characterise the multiaxial fatigue behaviour of the alloy. In these tests, the axial and torsional loads were applied simultaneously, with their peaks aligned in phase.
On the other hand, nonproportional loading conditions can involve applying axial and torsional loads that are out of phase, with their peaks occurring at different times. This results in the principal planes becoming a function of time and, thus, rotating. The Coffin-Manson equations, commonly used for modelling fatigue behaviour, are unable to account for these complex multiaxial loading conditions.
To address this limitation, Garud proposed an energy model based on plastic strain energy. This model is applicable to materials exhibiting Masing behaviour under multiaxial proportional axial-torsional loading. By considering the cyclic axial and shear strain hardening exponents, normal and shear stress ranges, and normal and shear plastic strain ranges, the plastic strain energy can be calculated for multiaxial axial-torsion load paths.
Furthermore, the Critical Plane Approach is another valuable tool for assessing damage from multiaxial loads. It focuses on the stresses and strains on critical planes, which can be maximum shear or tensile stress planes depending on the material and stress states. By employing this approach, separate damage models can be utilised for crack growth due to shear and tension, making it applicable to multiaxial axial-torsion load paths.
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Finite-element calculation of plasticity
Finite-element analysis (FE) is a powerful tool for calculating plasticity and has been used in a variety of applications, from modelling metal plasticity to calculating the applied J-integral for cracked ship structural components. The elastic-plastic finite-element method can be used to model metal deformation, taking into account both elastic and plastic deformation. This is particularly useful in hot working conditions where the elastic component of strain can be ignored.
Plasticity algorithms are often used in finite-element analysis programs to model non-linear material behaviour. Each new load increment increases stress levels, strain levels, and displacements. During each increment, the plasticity algorithm is used at the element level. The yield point and post-yield hardening describe the plastic behaviour of a material. The yield point is where the behaviour changes from elastic to plastic deformation. The yield stress may increase during plastic straining, and this is an important factor to consider in calculations.
The Von Mises yield criterion is a commonly used yield criterion in nonlinear material models. It is useful in cases of multi-axial loading as it provides a single equivalent scalar measure that can be compared with the yield stress of the material to predict whether it is yielding. The Tresca yield criterion is also defined in a similar way, with elastic deformation occurring for f (σij) < 0 and plastic deformation for f(σij) = 0.
When performing finite-element analysis, it is important to provide proper stress-strain data. The mechanical sublayer model is one of the most efficient plasticity models and has been shown to be effective in studying computational efficiencies. The rigid-plastic and viscoplastic finite-element techniques are useful approaches to modelling metal deformation when the elastic component of strain can be ignored or is minimal.
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Modelling as multilinear or bilinear
Multilinear modelling aims to replicate the material's behaviour as closely as possible to reality. In this method, the material's properties are established using a stress-strain curve obtained from a tensile test. The strain hardening curve is plotted using linear lines tangent to the nonlinear behaviour. This allows the model to follow a complicated stress-strain curve. Multilinear modelling is useful when dealing with complex models or when higher accuracy is required. However, it requires transforming the Engineering Stress-Strain curve data to True Stress-True Strain before inputting it into the model.
On the other hand, bilinear modelling provides a simpler and quicker approach. It is described by a stress-strain curve, where the Young modulus and the plastic phase modulus are used for material characterization. The plastic phase modulus should be non-negative and less than or equal to the Young modulus. When the plastic phase modulus is close to the Young modulus, the bilinear model simulates linear elastic analysis. On the other hand, when the plastic phase modulus is very small or zero, the model simulates perfect plasticity. Bilinear modelling is suitable when there is no complexity in geometries and loadings in the simulation.
Both multilinear and bilinear models are used in the finite-element calculation of plasticity, which can be time-consuming and data-intensive. These models are particularly useful in fatigue life estimation, where engineers can use simplified elastic behaviour to assess the durability of structures.
In summary, the choice between multilinear and bilinear modelling depends on the specific requirements of the analysis. Multilinear modelling provides a more accurate representation of the material's behaviour but is more complex and time-consuming. Bilinear modelling, on the other hand, offers a simpler and quicker approach that is suitable for less complex simulations.
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Plasticity under extreme conditions
Plasticity, in the context of stress and strain calculations, refers to the ability of a material to deform irreversibly without breaking. This is in contrast to elasticity, where a material returns to its original shape after the load is removed. Fully plastic stress refers to the state where the material has exceeded its yield point and exhibits permanent deformation.
Now, let's discuss plasticity under extreme conditions:
In biology, phenotypic plasticity refers to the ability of organisms to alter their traits or behaviors in response to changing environmental conditions. Extreme conditions in this context could be high temperatures, drought, or other environmental stresses. For example, the survival of plants under dry conditions is influenced by traits such as stomatal density, leaf shape, root length, and anatomy. Insects' responses to drought may involve changes in metabolic rate, size, and water storage abilities.
Empirical evidence suggests that species from more variable environments tend to exhibit higher levels of phenotypic plasticity, making them better equipped to survive extreme conditions. This is because they have already experienced a range of conditions and have had opportunities to develop plastic responses. However, plasticity can also be maladaptive under extreme environments if there is no strong genetic correlation between extreme and non-extreme states.
Additionally, in engineering, the finite-element calculation of plasticity under extreme conditions can be challenging and time-consuming due to the need for extensive data. As a result, engineers often use simplified methods that consider only elastic behavior or employ approximate methods that take into account both elastic and plastic behaviors for more accurate predictions.
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Frequently asked questions
The stress in plastic is calculated by dividing the force by the area.
The strain in plastic is calculated by dividing the change in length by the original length.
Stress = F / A.
