# Plasticity (physics)

 Continuum mechanics Key topics Conservation of massConservation of momentumNavier-Stokes equations Classical mechanics Stress Strain Tensor Solid mechanics Solids Elasticity Plasticity Hooke's lawRheology Viscoelasticity Fluid mechanics Fluids Fluid staticsFluid dynamics Viscosity Newtonian fluidsNon-Newtonian fluidsSurface tension Scientists Newton Stokes others This box:  •  • [ edit]
In physics and materials science, plasticity is a property of a material to undergo a non-reversible change of shape in response to an applied force. For example, a solid piece of metal or plastic being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself. By contrast, a permanent crease in a sheet of paper or a re-shaping of wet clay is due to a rearrangement of separate fibers or particles. In engineering, the transition from elastic behavior to plastic behavior is called yield.

## Explanation

For many ductile metals, tensile loading applied to a sample will cause it to behave in an elastic manner. Each increment of load is accompanied by a proportional increment in extension, and when the load is removed, the piece returns exactly to its original size. However, once the load exceeds some threshold (the yield strength), the extension increases more rapidly than in the elastic region, and when the load is removed, some amount of the extension remains. A generic graph displaying this behavior is below.

Plasticity is a property of materials to undergo large deformation without fracture. This is found in most metals, and in general is a good description of a large class of materials. Perfect plasticity is a property of materials to undergo large shear deformation without any increase of (shear) stress. Plastic materials that are not perfectly plastic are visco-plastic.

Microscopically, plasticity in metals is a consequence of dislocations.

## Mathematical descriptions of Plasticity

### Deformation theory

There are several mathematical descriptions of Plasticity. One is deformation theory (see e.g. Hooke's law) where the stress tensor (of order d in d dimensions) is a function of the strain tensor. Although this description is accurate when a small part of matter is subjected to increasing loading (such as strain loading), this theory cannot account for irreversibility.

The image above represents a shear stress component with respect to a shear strain component, under increasing strain loading.

Ductile materials can sustain large plastic deformations without fracture. However, even ductile metals will fracture when the strain becomes large enough - this is as a result of work-hardening of the material, which causes it to become brittle. Heat treatment such as annealing can restore the ductility of a worked piece, so that shaping can continue.

### Flow plasticity theory

In 1934, Egon Orowan, Michael Polanyi and Geoffrey Ingram Taylor, roughly simultaneously, realized that the plastic deformation of ductile materials could be explained in terms of the theory of dislocations. The more correct mathematical theory of plasticity, 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 of deformation.

## Elastic vs plastic failure

If the stress exceeds a critical value, as was mentioned above, the material will undergo plastic, or irreversible, deformation. This critical stress can be tensile or compressive.

### Tresca Criterion

This 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. Given the principal stress state, we can use Mohr’s Circle to solve for the maximum shear stresses our material will experience and conclude that the material will fail if:

σ1 - σ3 ≥ σ0

Where σ1 is the maximum normal stress, σ3 is the minimum normal stress, and σ0 is the stress under which the material fails in uniaxial loading. A yield surface may be constructed, which provides a visual representation of this concept. Inside of the yield surface, deformation is elastic. Outside of the surface, deformation is plastic. See Henri Tresca.

### Von Mises Criterion

This criterion is based on the Tresca criterion but takes into account the assumption that hydrostatic stresses do not contribute to material failure. Von Mises solves for an effective stress under uniaxial loading, subtracting out hydrostatic stresses, and claims that all effective stresses greater than that which causes material failure in uniaxial loading will result in plastic deformation.

σeffective² = 1/2 ((σ11 – σ22)² + (σ22 – σ33)² + (σ11 – σ33)²) + 3 (σ12² + σ13² + σ23²)

Again, a visual representation of the yield surface may be constructed using the above equation, which takes the shape of an ellipse. Inside the surface, materials undergo elastic deformation. Outside of the surface they undergo plastic deformation. See Von Mises stress

## Atomic Mechanisms

### Slip Systems

Crystalline materials contain uniform planes of atoms organized with long-range order. Planes may slip past each other along their close-packed directions, as is shown on the slip systems wiki page. The result is a permanent change of shape within the crystal and plastic deformation. The presence of dislocations increases the likelihood of planes slipping.

### Shear Banding

The presence of other defects within a crystal may entangle dislocations or otherwise prevent them from gliding. When this happens, plasticity is localized to particular regions in the material. For crystals, these regions of localized plasticity are called shear bands.

### Crazing

In amorphous materials, the discussion of “dislocations” is inapplicable, since the entire material lacks long range order. These materials can still undergo plastic deformation. Since amorphous materials, like polymers, are not well-ordered, they contain a large amount of free volume, or wasted space. Pulling these materials in tension opens up these regions and can give materials a hazy appearance. This haziness is the result of crazing, where fibrils are formed within the material in regions of high hydrostatic stress. The material may go from an ordered appearance to a "crazy" pattern of strain and stretch marks.

## Martensitic materials

Some materials, especially those prone to Martensitic transformations, deform in ways that are not well described by the classic theories of plasticity and elasticity. One of the best-known examples of this is nitinol, which exhibits pseudoelasticity: deformations which are reversible in the context of mechanical design, but irreversible in terms of thermodynamics.

## Cellular materials

These materials plastically deform when the bending moment exceeds the fully plastic moment. This applies to open cell foams where the bending moment is exerted on the cell walls. The foams can be made of any material with a plastic yield point which includes rigid polymers and metals. This method of modeling the foam as beams is only valid if the ratio of the density of the foam to the density of the mater is less than 0.3. This is because beams yield axially instead of bending. In closed cell foams, the yield strength is increased if the material is under tension because of the membrane that spans the face of the cells.

## References

• R. Hill, The Mathematical Theory of Plasticity, Oxford University Press (1998).
• Jacob Lubliner, Plasticity Theory, Macmillan Publishing, New York (1990).
• L. M. Kachanov, Fundamentals of the Theory of Plasticity, Dover Books.
• A.S. Khan and S. Huang, Continuum Theory of Plasticity, Wiley (1995).
• J. C. Simo, T. J. Hughes, Computational Inelasticity, Springer.
• M. F. Ashby. Plastic Deformation of Cellular Materials. Encyclopedia of Materials: Science and Technology, Elsevier, Oxford, 2001, Pages 7068-7071.
• Van Vliet, K. J., 3.032 Mechanical Behavior of Materials, MIT (2006)
Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i.e., liquids and gases).

The fact that matter is made of atoms and that it commonly has some sort of heterogeneous
The law of conservation of mass/matter, also known as law of mass/matter conservation (or the Lomonosov-Lavoisier law), states that the mass of a closed system will remain constant, regardless of the processes acting inside the system.
The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances such as liquids and gases. These equations establish that changes in momentum in infinitesimal volumes of fluid are simply the sum of dissipative
Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies.
Stress is a measure of force per unit area within a body. It is a body's internal distribution of force per area that reacts to external applied loads. Stress is often broken down into its shear and normal components as these have unique physical significance.
strain is the geometrical expression of deformation caused by the action of stress on a physical body. Strain is calculated by first assuming a change between two body states: the beginning state and the final state.
The term tensor has slightly different meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is a multilinear function.
Solid mechanics is the branch of physics and mathematics that concerns the behavior of solid matter under external actions (e.g., external forces, temperature changes, applied displacements, etc.). It is part of a broader study known as continuum mechanics.
A solid object is in the states of matter characterized by resistance to deformation and changes of volume. At the microscopic scale, a solid has these properties :
• The atoms or molecules that comprise the solid are packed closely together.

Elasticity is a branch of physics which studies the properties of elastic materials. A material is said to be elastic if it deforms under stress (e.g., external forces), but then returns to its original shape when the stress is removed.
Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress).
Rheology is the study of the deformation and flow of matter under the influence of an applied stress, which might be shear stress or extensional stress. Rheology dealing with shear stress is called shear rheology.
Viscoelasticity, also known as anelasticity, is the study of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied.
Fluid mechanics is the study of how fluids move and the forces on them. (Fluids include liquids and gases.) Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of fluids in motion.
FLUID (Fast Light User Interface Designer) is a graphical editor that is used to produce FLTK source code. FLUID edits and saves its state in text .fl files, which can be edited in a text editor for finer control over display and behavior.
Fluid statics (also called hydrostatics) is the science of fluids at rest, and is a sub-field within fluid mechanics. The term usually refers to the mathematical treatment of the subject.
Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. It has several subdisciplines itself, including aerodynamics (the study of gases in motion) and hydrodynamics (the study of liquids in motion).
Viscosity is a measure of the resistance of a fluid to deform under either shear stress or extensional stress. It is commonly perceived as "thickness", or resistance to flow.
A Newtonian fluid (named for Isaac Newton) is a fluid that flows like water—its stress versus rate of strain curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.
A non-Newtonian fluid is a fluid in which the viscosity changes with the applied strain rate. As a result, non-Newtonian fluids may not have a well-defined viscosity.
Surface tension is an effect within the surface layer of a liquid that causes that layer to behave as an elastic sheet. It allows insects, such as the water strider (pond skater, UK), to walk on water.
Sir Isaac Newton

Isaac Newton at 46 in
Godfrey Kneller's 1689 portrait
Born 4 January 1643 [OS: 25 December 1642]
George Stokes

Sir George Gabriel Stokes, 1st Baronet
Born 13 July 1819
Skreen, County Sligo, Ireland
Plastic is the general term for a wide range of synthetic or semisynthetic polymerization products. They are composed of organic condensation or addition polymers and may contain other substances to improve performance or economics.
Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
Materials science or materials engineering is an interdisciplinary field involving the properties of matter and its applications to various areas of science and engineering. This science investigates the relationship between the structure of materials and their properties.
yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to plastically deform. Prior to the yield point the material will deform elastically and will return to its original shape when the applied