Omar Khalid
Plastic flow is a solid mechanics theory that is used to describe the plastic behaviour of materials. It is a branch of physics known as rheology, which studies the deformation and flow of materials under stress. When a load is applied, the material behaves like a flowing fluid and continues to do so as long as the forces are applied. This deformation is called plastic deformation, and the material is called elastic-plastic. The plastic flow rule and evolution equations are given by the kinematic hardening modulus and the internal variables are accumulated effective plastic stress.
| Characteristics | Values |
|---|---|
| Definition | Deformation of a material that remains rigid under stresses of less than a certain intensity but behaves under severer stresses approximately as a Newtonian fluid. |
| Other names | Plastic deformation, creep |
| Materials | Rocks, ice, metals |
| Processes | Intracrystalline gliding, recrystallization, rolling, pressing, forging, rock folding, rock flow |
| Formula | The plastic flow rule and evolution equations are given by ... κ (the kinematic hardening modulus) and the internal variables are accumulated effective plastic stress |
| Plasticity type | Rate-independent plasticity, rate-dependent plasticity |
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What You'll Learn
Plastic deformation
The rate of plastic deformation is initially high but eventually tapers off to a steady value. This steady value, known as the shear-strain rate, can be plotted against the applied stress to create a curved graph. The deformation speed also plays a role in plastic deformation, with higher stresses typically required to increase the rate of deformation. This behaviour is described as viscoplasticity, and it is observed in materials that exhibit a time-dependent deformation response, where the degree of deformation increases with prolonged exposure to stress.
The study of plastic deformation involves various theories and models, such as the J2 flow theory and the hypoelastic-plastic constitutive model. These models incorporate parameters such as stress, strain, and viscosity to describe the behaviour of materials under different conditions. By studying plastic deformation, scientists and engineers can make informed decisions about material selection, design considerations, and the prediction of material behaviour in various applications.
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Rheology
Plastic flow is of particular interest in the study of glacier flow, where it involves two processes: intracrystalline gliding and recrystallization. In intracrystalline gliding, the layers within an ice crystal shear parallel to each other without disrupting the crystal lattice, allowing the ice to flow around and over obstacles. In recrystallization, the crystal lattice is disrupted, leading to a change in the microstructure of the ice.
Plastic flow is also observed in metal-forming processes such as rolling, pressing, and forging, as well as in geologic processes such as rock folding and rock flow within the Earth under extremely high pressures and temperatures. In these cases, the plastic flow is described by flow plasticity theories, which assume that the total strain in a body can be decomposed into an elastic part and a plastic part. The elastic part of the strain can be computed from a linear elastic or hyperelastic model, while the plastic part requires a flow rule and a hardening model.
The flow rule describes the relationship between the stress and strain in the material, and it is used to determine the amount of plastic deformation. The hardening model describes how the material becomes stronger or harder as a result of the plastic deformation. These models are often based on Kirchhoff stress formulations of J2 flow theory plasticity, which take into account the additive decomposition of the rate-of-deformation tensor into elastic and plastic parts.
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Hypoelastic-plastic constitutive model
Plasticity theory was developed to predict the behaviour of metals under loads exceeding the plastic range. However, the theory has since been adapted to other materials, including polymers and some types of soil.
The theory of plasticity differs from the theory of elasticity in the relationship between stress and strain. The theory of elasticity is described by the generalized Hooke law. On the other hand, the theory of plasticity is based on Drucker's postulate, as well as the loading and unloading criteria, with an emphasis on the incremental theory (flow theory).
In this context, hypoelastic-plastic constitutive models are used when elastic strains are small compared to plastic strains. These models assume an additive decomposition of the rate-of-deformation tensor into elastic and plastic parts. The elastic response is hypoelastic, and the choice of objective stress rate depends on several factors. One example is the Jaumann rate of Cauchy stress.
A novel approach is to use the principal axes technique to develop a new hypoelastic constitutive model for an isotropic elastic solid in finite deformation. This model is combined with an isotropic flow rule to form an elastic-plastic rate constitutive equation. The use of the principal axes technique ensures that the stress tensor is coaxial with the elastic stretch tensor and that solutions are independent of the choice of objective stress rate.
In the context of soft tissues, an isotropic, isochoric, hypoelastic, constitutive model has been applied to in-plane biaxial experiments. This model provides a reasonably good description of the non-linear normal responses observed in experimental data. However, there is room for improvement in predicting the shear response.
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Flow plasticity theories
Plastic flow refers to the deformation and flow of materials under stress, beyond their yield point. This phenomenon is studied under the branch of physics known as rheology. When a material is in motion, its molecules and larger particles experience forces that cause them to slide along each other, generating resistance from internal molecular or particle friction. This resistance is what we perceive as viscosity.
The J2 flow theory is a commonly referenced model in flow plasticity. It involves a hypoelastic-plastic constitutive model with combined isotropic kinematic hardening. The plastic flow equations and tangent modulus are derived from the kinematic hardening modulus (κ) and the accumulated effective plastic stress as internal variables. Johnson and Bammann (1984) contributed significantly to this theory by demonstrating the elimination of nonphysical oscillations through the Green-Naghdi rate of stress and back stress.
Another important concept in flow plasticity theories is the plastic flow rule. This rule assumes that the plastic strain increment and deviatoric stress tensor share the same principal directions. However, in rock plasticity theories, this assumption is relaxed due to the pressure-dependence of the yield surface. Instead, it is assumed that the plastic strain increment and the normal to the pressure-dependent yield surface are co-directional. This co-directionality is known as the normality condition.
The Prager consistency condition is employed to validate the associated flow rule and close the set of constitutive equations. Large deformation flow theories of plasticity often begin with two key assumptions: the additive decomposition of the rate-of-deformation tensor into elastic and plastic parts, or the multiplicative decomposition of the deformation gradient tensor into similar components. While the former assumption was initially prevalent in numerical simulations of metals, it has been gradually superseded by the multiplicative theory.
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Drucker's second stability postulate
Plastic flow refers to the deformation and flow of materials under stress, specifically, how solids start to flow after reaching a yield point. This phenomenon is studied in the branch of physics known as rheology.
Drucker's stability postulates refer to a set of mathematical criteria that restrict the possible nonlinear stress-strain relations that can be satisfied by a solid material. Drucker's second stability postulate states that a material that does not satisfy these criteria will be unstable, meaning that applying a load to a material point can lead to arbitrary deformations at that point unless additional conditions are met.
In simpler terms, this postulate predicts that materials that do not meet the criteria will be unable to withstand stress without deforming. This postulate is particularly relevant in the context of plastic flow because it helps identify materials that are stable and well-suited for numerical analysis, as well as those that may present difficulties due to their instability.
While Drucker's postulate is a sufficient condition for plastic stability, it is not necessary, as evidenced by experimental observations of metal alloys that violate the associated flow rule (AFR). This has led to the proposal of alternative material models based on non-associated flow, which can ensure stability while providing potentially more accurate representations of material behaviour.
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Frequently asked questions
Plastic flow is a solid mechanics theory that describes the plastic behaviour of materials. It is a type of deformation that occurs when a material flows as a result of an applied load and remains deformed even after the load is removed.
Plastic flow occurs in many metal-forming processes such as rolling, pressing, and forging. It is also observed in geological processes like rock folding and rock flow within the Earth under extremely high pressures and elevated temperatures. In the case of ice, plastic flow happens due to pressure at depth, and a thickness of around 22 meters is required for plastic flow in temperate glaciers.
Elastic deformation refers to the recoverable part of the deformation, where the material returns to its original shape when the load is removed. On the other hand, plastic deformation is the unrecoverable part, where the material retains its deformed shape even after the load is removed.
Plastic flow is modelled using flow plasticity theories, which assume that the total strain in a body can be decomposed into an elastic part and a plastic part. The elastic part can be computed using a linear elastic or hyperelastic constitutive model. However, determining the plastic part requires a flow rule and a hardening model. The flow rule describes the relationship between stress and strain, and the hardening model accounts for the accumulation of plastic strain over time.
