Maya Ellison
Plasticity, also known as plastic deformation, is the ability of a solid material to undergo irreversible deformation, a permanent change of shape in response to applied forces. The mathematical theory of plasticity, or flow plasticity theory, uses a set of non-linear, non-integrable equations to describe the set of changes on strain and stress with respect to a previous state and a small increase in deformation. There are several mathematical descriptions of plasticity, including deformation theory, and the theory of dislocations, which explains the plastic deformation of ductile materials.
| Characteristics | Values |
|---|---|
| Definition | Plasticity is the ability of a solid material to undergo permanent deformation, a non-reversible change of shape in response to applied forces. |
| Other Names | Plastic deformation, plastic flow |
| Application | Plasticity is observed in most materials, particularly metals, soils, rocks, concrete, and foams. |
| Mathematical Theories | Flow plasticity theory, deformation theory, continuum theory, total deformation theory, friction theory |
| Yield Criteria | Tresca criterion, von Mises criterion |
| Yield Functions | Prager consistency condition, Drucker stability postulate |
| Plastic Deformation Causes | Slip at microcracks, bubble or cell rearrangements, dislocations, slip and twinning |
| Plasticity in Crystals | Caused by slip and twinning deformations in the crystal lattice |
| Plasticity in Brittle Materials | Caused predominantly by slip at microcracks |
| Plasticity in Cellular Materials | Caused by bubble or cell rearrangements |
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What You'll Learn
Flow plasticity theory
Plasticity is the ability of a solid material to undergo permanent deformation, a non-reversible change of shape in response to applied forces. This is also known as plastic deformation and is observed in most materials, particularly metals, soils, rocks, concrete, and foams.
The mathematical theory of plasticity, or flow plasticity theory, uses a set of non-linear, non-integrable equations to describe the set of changes on strain and stress with respect to a previous state and a small increase in deformation. This theory is used to describe the "plastic flow" of a material.
In metal plasticity, the flow rule assumes that the plastic strain increment and deviatoric stress tensor have the same principal directions. Rock plasticity theories use a similar concept, except that the pressure-dependence of the yield surface must be assumed. This form of the flow rule is called an associated flow rule, and the assumption of co-directionality is called the normality condition.
The Prager consistency condition is needed to close the set of constitutive equations and eliminate the unknown parameters. Large deformation flow theories of plasticity typically start with one of two assumptions: the rate of deformation tensor can be decomposed into elastic and plastic parts, or the deformation gradient tensor can be multiplicatively decomposed into elastic and plastic parts.
The flow theory is more widely used in the description of phenomena and is more popular because it directly describes the transition between elastic and inelastic states.
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Deformation theory
In deformation theory, the Cauchy stress tensor (of order d-1 in d dimensions) is a function of the strain tensor. This description is accurate when a small part of a material is subjected to increasing loading, such as strain loading. However, deformation theory cannot account for irreversibility.
Ductile materials can sustain large plastic deformations without fracture. However, they will fracture when the strain becomes large enough due to work hardening, which causes the material to become brittle. Heat treatment can restore the ductility of a worked piece, allowing shaping to continue.
In microscopic theory, plasticity is interpreted as deformation by dislocation processes, whereas in macroscopic continuum mechanics, it refers to any type of permanent deformation, especially in cases where time or rate of deformation effects are not dominant.
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Plastic deformation
In crystalline materials, plasticity is typically associated with dislocations, which are defects in the crystal lattice that allow for slip and twinning deformations. Ductile materials, such as metals, can sustain large plastic deformations without fracture. However, even these materials have their limits, and they will eventually fracture when subjected to sufficiently high strain. This is due to work hardening, which causes the material to become brittle.
In contrast, brittle materials like rock, concrete, and bone exhibit plasticity predominantly through slip at microcracks. On the other hand, cellular materials like liquid foams or biological tissues display plasticity mainly due to bubble or cell rearrangements.
The mathematical theory of plasticity, or flow plasticity theory, employs a set of non-linear, non-integrable equations to describe the changes in strain and stress concerning a previous state and a small increase in deformation. This theory distinguishes between elastic and plastic deformations, with the latter being irreversible. The critical stress at which a material undergoes plastic deformation can be tensile or compressive, and various criteria, such as the Tresca and von Mises criteria, are used to determine material yielding.
Plasticity theory encompasses several approaches, including continuum theory, crystallographic mechanisms, and total deformation theory. Continuum theory focuses on yield criteria and predicting the stress states that lead to yielding. Crystallographic mechanisms, on the other hand, examine slip and twinning deformations at the crystallographic level. Total deformation theory relates stresses to total strain, providing better predictions for large deformation phenomena like buckling in elasto-plastic materials.
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Perfect plasticity
The concept of perfect plasticity is associated with the mechanical behaviour of materials. The yield function of a perfectly plastic material is only a function of the stress state. The elastic domain characterising the mechanical behaviour of such materials is always the same. The simplest mathematical expression of the yield surface for a perfectly plastic material is expressed in terms of the relevant stress (total or effective stress).
The theory of perfect plasticity was initially used to solve geotechnical stability problems involving geomaterials by Coulomb (1773) and Rankine (1857). It has since been applied to the analysis of other materials such as metals by Saint Venant (1870) and Lévy (1870). Prager (1949) proposed that when a material characterised by plastic behaviour is loaded, the stress state must remain on the yield surface. This is known as the condition of consistency, which is necessary to ensure an appropriate description of the physical process involved in plastic deformation.
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Yield criteria
The mathematical theory of plasticity, or flow plasticity theory, uses a set of non-linear, non-integrable equations to describe the set of changes on strain and stress with respect to a previous state and a small increase in deformation. If the stress exceeds a critical value, the material will undergo plastic or irreversible deformation. This critical stress can be tensile or compressive.
The Tresca and von Mises criteria are commonly used to determine whether a material has yielded. However, these criteria have proved inadequate for a large range of materials and several other yield criteria are also in widespread use. The Tresca criterion is based on the notion that when a material fails, it does so in shear, which is a relatively good assumption when considering metals. The von Mises criterion is also known as the "maximum distortion energy criterion". It states that a material will yield when the distortion energy due to shear reaches a critical value. This criterion is based on the idea that the yield strength of a material is proportional to its shear modulus.
There are several other yield criteria that have been proposed, including the Coulomb criterion, the Drucker-Prager criterion, and the Mohr-Coulomb criterion. The Coulomb criterion is based on the idea that the yield strength of a material is proportional to the tangent of the angle of internal friction of the material. The Drucker-Prager criterion is a modification of the Coulomb criterion that includes the effect of confinement pressure. The Mohr-Coulomb criterion is a three-dimensional generalization of the Coulomb criterion that includes the effect of the orientation of the stress axis.
In addition to these criteria, there are also more advanced plasticity theories that consider the evolution of microstructure during deformation. These theories are often based on the concept of flow rules, which describe how the material will deform given a certain stress state. The associated flow rule is obtained when the yield and plastic potential functions are identical. In general, when these functions are not equal, the flow rule is nonassociative.
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Frequently asked questions
Plasticity theory is the mathematical theory used to describe the "plastic flow" of a material. It deals with the yielding of materials, often under complex states of stress.
Plastic deformation is the ability of a solid material to undergo permanent deformation, a non-reversible change of shape in response to applied forces. Unlike elastic deformation, plastic deformations are usually instantaneous and permanent.
A solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself.
The mathematical theory of plasticity, or flow plasticity theory, uses a set of non-linear, non-integrable equations to describe the set of changes on strain and stress with respect to a previous state and a small increase in deformation.
Perfect plasticity is a property of materials to undergo irreversible deformation without any increase in stresses or loads.
