# strain (materials science)

 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 any branch of science dealing with materials and their behaviour, 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. Then the difference in placement of two points in this body in those two states expresses the numerical value of strain. Strain therefore expresses itself as a change in size and/or shape.

If strain is equal over all parts of a body, it is referred to as homogeneous strain; otherwise, it is inhomogeneous strain. In its most general form, the strain is a symmetric tensor.

## Quantifying strain

Given that strain results in the deformation of a body, it can be measured by calculating the change in length of a line or by the change in angle between two lines (where these lines are theoretical constructs within the deformed body). The change in length of a line is termed the stretch, absolute strain, or extension, and may be written as . Then the (relative) strain, , is given by

where
is strain in measured direction
is the original length of the material.
is the current length of the material.

The extension () is positive if the material has gained length (in tension) and negative if it has reduced length (in compression). Because is always positive, the sign of the strain is always the same as the sign of the extension.

Strain is a dimensionless quantity. It has no units of measure because in the formula the units of length "cancel out".

Strain is often expressed in dimensions of metres/metre or inches/inch anyway, as a reminder that the number represents a change of length. But the units of length are redundant in such expressions, because they cancel out. When the units of length are left off, strain is seen to be a pure number, which can be expressed as a decimal fraction, a percentage or in parts-per notation. In common solid materials, the change in length is generally a very small fraction of the length, so strain tends to be a very small number. It is very common to express strain in units of micrometre/metre or μm/m. When the units of μm/m are canceled out, strain is expressed as a number followed by μ, the SI prefix all by itself. It is usually clear from the context that μ is used for its SI prefix meaning, which is interchangeable with "x 10−6" or "ppm" (parts per million), and not one of the many other possible meanings for μ.

## Linear axial strain at single point

In the case of measuring strain in the selected point of the body, it is expressed as a strain where the distance between two points approaches zero:

where
is strain in measured direction
is the length difference for current length .
is the current length of the material, which approaches zero.

Therefore linear strain is defined as change of distance in the close proximity of selected point.

## The general case of linear strain

For the body of any shape, subjected to any deformation the values of strain will be different depending on the spatial direction of measurement. Considering the linear deformation in the point A placed at the start of coordinate system and a second point B placed along the x axis, which due to deformation has moved to the point B' the linear strain will be expressed as:

Doing similar calculations for axes y and z respective values of εy and εz can be obtained. For any given displacement field (the values of displacement vectors for all points in the body) the linear strain can be written as:

: ; ;

where
is strain in direction along axis i
is a differential of at any point in the direction along axis i

## Shear strain

Similarly the angular change at any point between two lines crossing this point in a body can be measured as a shear (or shape) strain. Shear strain γ is the limit of ratio of angular difference between any two lines in a body before and after deformation, assuming that the lines lengths are approaching zero. Given a displacement field like above, the shear strain can be written as follows:

; ;

## Volumetric strain

Although linear strain ε and shear strain γ completely defines the state of deformation of a body, it is also possible to measure other characteristic strain values, like for example volumetric strain, which measures the ratio of change of body's volume. The definition of volumetric strain at selected point is:

where
is volumetric strain
is initial volume
is final volume

For cartesian coordinate systems, the following expression is a first order approximation:

where
is volumetric strain
are strains along x, y and z axis

## The strain tensor

Main article: Strain tensor

Using above notation for linear and shear strain it is possible to express strain as a strain tensor:

using indicial notation or using vector notation:

Comparing traditional notation with tensor notation following is obtained for cartesian coordinate system:

Then volumetric strain equals:

where gij is a contravariant metric tensor (using tensor notation: )

## Principal strains in two dimensions

Because the strain tensor is a real symmetric matrix, by singular value decomposition it can be represented as a set of orthogonal eigenvectors, directions along which there is no shear, only stretching or compression.

Assuming the two dimensional strain tensor given as:

Then principal strains are equal to the eigenvalues of :

## The case of large deformations

Above reasoning assumes that body is subject to small deformations. It must be rememberred that with increasing deformation the linear strain error increases. For large deformations the strain tensor can be written as:

where

gij is the metric tensor of body after deformation

gij(0) is metric tensor of the undeformed body

## Engineering strain vs. true strain

In the definition of linear strain (known technically as engineering strain), strains cannot be totaled. Imagine that a body is deformed twice, first by and then by (cumulative deformation). The final strain
is slightly different from the sum of the strains:
and
As long as , it is possible to write:
and thus

True strain (aka natural strain and logarithmic strain and Hencky's strain), however, can be totaled. This is defined by:
and thus
where
is the original length of the material.
is the final length of the material.
The engineering strain formula is the series expansion of the true strain formula.

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.
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.
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.
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
In engineering mechanics, deformation is a change in shape due to an applied force. This can be a result of tensile (pulling) forces, compressive (pushing) forces, shear, bending or torsion (twisting). Deformation is often described in terms of strain.
Strain can refer to:
• Strain (materials science), the deformation of materials caused by stress
• Strain (biology), a variant of a plant, virus or bacterium; or an inbred animal used for experimental purposes
• Strain (injury), a muscle injury
• Strain (music)

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.
The strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation:
• the diagonal coefficients εii are the relative change in length in the direction of the i

angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept
Tension is a reaction force applied by a stretched string (rope or a similar object) on the objects which stretch it. The direction of the force of tension is parallel to the string, towards the string.
Physical compression is the result of the subjection of a material to compressive stress, resulting in reduction of volume. The opposite of compression is tension.