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## Strain (materials science)
In any branch of science dealing with materials and their behaviour,
If strain is equal over all parts of a body, it is referred to as
## Quantifying strainGiven 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, ε, 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 "× 10 ## Linear axial strain at single pointIn 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.
## The general case of linear strainFor 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 placed along the B axis, which due to deformation has moved to the point x the linear strain will be expressed as:
B' Doing similar calculations for axes respective values of z and ε_{y} 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:
ε_{z}- ; ;
where - is strain in direction along axis
*i* - is a differential of at any point in the direction along axis
*i*
## Shear strainSimilarly 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 strainAlthough linear strain where - is volumetric strain
*V*^{(0)}is initial volume*V*is final volume
For cartesian coordinate systems, the following expression is a first order approximation: where - is volumetric strain
- are strains along
,**x**and**y**axis**z**
## The strain tensorUsing 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 where
## Principal strains in two dimensionsBecause 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 deformationsAbove reasoning assumes that body is subject to where
## Engineering strain vs. true strainIn the definition of linear strain (known technically as is slightly different from the sum of the strains: and As long as , it is possible to write: and thus
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. ## See also- Elongation (materials science) or Stretch ratio
- Stress
- Strain gauge
- Strain tensor
- Stress-strain curve
- Stretch ratio
- Hooke's law
- Poisson's ratio
- Finite deformation tensors
- Strain Rate
Categories: Continuum mechanics | Materials science |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Strain_(materials_science)". A list of authors is available in Wikipedia. |