Strain (materials science)

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This article is about the deformation of materials. For other uses, see strain.

Strain is the deformation of materials caused by the action of stress. 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.

1 Quantifying strain
2 Linear axial strain at single point
3 The general case of linear strain
4 Shear strain
5 Volumetric strain
6 The strain tensor
7 Principal strains in two dimensions
8 The case of large deformations
9 Compatibility equations
10 Engineering strain vs. true strain
11 Plane strain
12 See also
13 External links

[edit] 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 $\delta \ell$ . Then the (relative) strain, $ε$ , is given by

$\varepsilon = \frac {\delta \ell}{\ell_o} = \frac {\ell - \ell_o}{\ell_o}$

where

$\varepsilon$ is strain in measured direction

$\ell_o$ is the original length of the material

$\ell$ is the current length of the material

The extension ( $\delta \ell$ ) is positive if the material has gained length (in tension) and negative if it has reduced length (in compression). Because $\ell_o$ 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 meters/meter 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 micrometers/meter 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 "× 10⁻⁶" or "ppm" (parts per million), and not one of the many other possible meanings for μ.

[edit] 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 $\ell$ between two points approaches zero:

$\varepsilon = \mathop {\lim_{\ell \to 0}} \frac {{\delta} {\ell} } {\ell}$

where

$\varepsilon$ is strain in measured direction

${{\delta} {\ell} }$ is the length difference for current length $\ell$ .

$\ell$ 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.

[edit] 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:

$\varepsilon_x = \mathop {\lim_{B \to A}}{{|AB'|-|AB|} \over {|AB|}}$

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

$\varepsilon_x = {{\partial u_x} \over {\partial x}}$ ; $\varepsilon_y = {{\partial u_y} \over {\partial y}}$ ; $\varepsilon_z = {{\partial u_z} \over {\partial z}}$

where

$\varepsilon_i$ is strain in direction along axis i

${{\partial u_i} \over {\partial i}}$ is a differential or partial derivative of $\overrightarrow u$ at any point in the direction along axis i

[edit] 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 $\overrightarrow u$ like above, the shear strain can be written as follows:

$\gamma_{xy} = {{\partial u_x} \over {\partial y}} + {{\partial u_y} \over {\partial x}}$ ; $\gamma_{yz} = {{\partial u_y} \over {\partial z}} + {{\partial u_z} \over {\partial y}}$ ; $\gamma_{xz} = {{\partial u_x} \over {\partial z}} + {{\partial u_z} \over {\partial x}}$

[edit] Volumetric strain

Although linear strain ε and shear strain γ completely define 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:

$\vartheta = \lim_{V^{(0)} \to 0}{V - V^{(0)} \over {V^{(0)}}}$

where

$\vartheta$ is volumetric strain

V (0)

is initial volume

V

is final volume

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

$\vartheta = \varepsilon_x + \varepsilon_y + \varepsilon_z$

where

$\vartheta$ is volumetric strain

$\varepsilon_x , \varepsilon_y , \varepsilon_z$ are strains along x, y and z axis

[edit] 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:

$\varepsilon_{ij} = {1 \over 2} \left({\nabla_i u_j + \nabla_j u_i}\right)$

using indicial notation or using vector notation:

$\varepsilon = {1 \over 2} ( \vec{\nabla}\vec{u} + (\vec{\nabla}\vec{u})^T)$

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

$\varepsilon_{ij}= \left[{\begin{matrix} {\varepsilon _x } & \frac {\gamma _{xy} } {2} & \frac {\gamma _{xz} } {2} \\ \frac {\gamma _{yx} } {2} & {\varepsilon _y } & \frac {\gamma _{yz} } {2} \\ \frac {\gamma _{zx} } {2} & \frac {\gamma _{zy} } {2} & {\varepsilon _z } \end{matrix}}\right]$

Then volumetric strain equals:

$\vartheta = \varepsilon_{ij}g^{ij}$

where g^ij is a contravariant metric tensor (using tensor notation: $\vartheta = tr(\varepsilon)$ )

[edit] 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:

$\varepsilon_{ij}= \left[{\begin{matrix} {\varepsilon _x } & {\frac {\gamma _{xy}} {2}} \\ {\frac {\gamma _{xy}} {2}} & {\varepsilon _y } \\ \end{matrix}}\right]$

Then principal strains $\varepsilon _1, \varepsilon _2$ are equal to the eigenvalues of $\varepsilon$ :

$\varepsilon _1 = \frac {\varepsilon _x + \varepsilon _ y}{2} + \sqrt{ \left( \frac {\varepsilon _x - \varepsilon _y}{2} \right)^2 + \left( \frac{\gamma _{xy}} {2}\right)^2 }$

$\varepsilon _2 = \frac {\varepsilon _x + \varepsilon _ y}{2} - \sqrt{ \left( \frac {\varepsilon _x - \varepsilon _y}{2} \right)^2 + \left( \frac{\gamma _{xy}} {2}\right)^2 }$

[edit] 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:

$\varepsilon_{ij} = {1 \over 2}({g_{ij}-g_{ij}^{(0)}})$

where

g_ij is the metric tensor of body after deformation

g_ij⁽⁰⁾ is metric tensor of the undeformed body

[edit] Compatibility equations

For prescribed strain components $\ \varepsilon_{ij}$ the strain tensor equation $\ u_{i,j}+u_{j,i}= 2 \varepsilon_{ij}$ represents a system of six differential equations for the determination of three displacements components $\ u_i$ , giving an over-determined system. Thus, a solution does not generally exist for an arbitrary choice of strain components. Therefore, some restrictions, named compatibility equations, are imposed upon the strain components. With the addition of the three compatibility equations the number of independent equations is reduced to three, matching the number of unknown displacement components. These constraints on the strain tensor were discovered by Saint Venant, and are called the "Saint Venant compatibility equations".

The compatibility functions serve to assure a single-valued continuous displacement function $\ u_i$ . If the elastic medium is visualized as a set of infinitesimal cubes in the unstrained state, after the medium is strained, an arbitrary strain tensor may not yield a situation in which the distorted cubes still fit together without overlapping.

In index notation, the compatibility equations are expressed as

$\ \varepsilon_{ij,km}+\varepsilon_{km,ij}-\varepsilon_{ik,jm}-\varepsilon_{jm,ik}=0$

Engineering notation
$\ \frac{\partial^2 \epsilon_x}{\partial y^2} + \frac{\partial^2 \epsilon_y}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{xy}}{\partial x \partial y}$ $\ \frac{\partial^2 \epsilon_y}{\partial z^2} + \frac{\partial^2 \epsilon_z}{\partial y^2} = 2 \frac{\partial^2 \epsilon_{yz}}{\partial y \partial z}$ $\ \frac{\partial^2 \epsilon_x}{\partial z^2} + \frac{\partial^2 \epsilon_z}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{zx}}{\partial z \partial x}$ $\ \frac{\partial^2 \epsilon_x}{\partial y \partial z} = \frac{\partial}{\partial x} \left ( -\frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z}\right)$ $\ \frac{\partial^2 \epsilon_y}{\partial z \partial x} = \frac{\partial}{\partial y} \left ( \frac{\partial \epsilon_{yz}}{\partial x} - \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z}\right)$ $\ \frac{\partial^2 \epsilon_z}{\partial x \partial y} = \frac{\partial}{\partial z} \left ( \frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} - \frac{\partial \epsilon_{xy}}{\partial z}\right)$

Engineering notation

$\ \frac{\partial^2 \epsilon_x}{\partial y^2} + \frac{\partial^2 \epsilon_y}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{xy}}{\partial x \partial y}$

$\ \frac{\partial^2 \epsilon_y}{\partial z^2} + \frac{\partial^2 \epsilon_z}{\partial y^2} = 2 \frac{\partial^2 \epsilon_{yz}}{\partial y \partial z}$

$\ \frac{\partial^2 \epsilon_x}{\partial z^2} + \frac{\partial^2 \epsilon_z}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{zx}}{\partial z \partial x}$

$\ \frac{\partial^2 \epsilon_x}{\partial y \partial z} = \frac{\partial}{\partial x} \left ( -\frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z}\right)$

$\ \frac{\partial^2 \epsilon_y}{\partial z \partial x} = \frac{\partial}{\partial y} \left ( \frac{\partial \epsilon_{yz}}{\partial x} - \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z}\right)$

$\ \frac{\partial^2 \epsilon_z}{\partial x \partial y} = \frac{\partial}{\partial z} \left ( \frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} - \frac{\partial \epsilon_{xy}}{\partial z}\right)$

[edit] 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 $\delta \ell_1$ and then by $\delta \ell_2$ (cumulative deformation). The final strain

$\epsilon = \frac{\delta \ell_1 + \delta \ell_2}{\ell_0}$

is slightly different from the sum of the strains:

$\epsilon_1 = \frac{\delta \ell_1}{\ell_0}$

and

$\epsilon_2 = \frac{\delta \ell_2}{\ell_0 + \delta \ell_1}$

As long as $\delta \ell_1 \ll \ell_0$ , it is possible to write:

$\epsilon_2 \simeq \frac{\delta \ell_2}{\ell_0}$

and thus

$\epsilon \simeq \epsilon_1 \; + \epsilon_2 \;$

True strain (also known as natural strain and logarithmic strain and Hencky's strain), however, can be totaled. This is defined by:

$\exp(\epsilon _{T}) = \frac{\ell_f}{\ell_0}$

and thus

$\epsilon _{T}= \ln \left (\frac{\ell_f}{\ell_0} \right )$

where

$\ell_0$ is the original length of the material.

$\ell_f$ is the final length of the material.

The engineering strain formula is the series expansion of the true strain formula.

[edit] Plane strain

In real engineering components, stress (and strain) are 3-D tensors but in prismatic structures such as a long metal billet, the length of the structure is much greater than the other two dimensions. The strains associated with length, i.e the normal strain $ε 33$ and the shear strains $ε 13$ and $ε 23$ (if the length is the 3-direction) are constrained by nearby material and are small compared to the cross-sectional strains. The strain tensor can then be approximated by:

$\underline{\underline{\epsilon}} = \begin{bmatrix} \epsilon_{11} & \epsilon_{12} & 0 \\ \epsilon_{21} & \epsilon_{22} & 0 \\ 0 & 0 & 0\end{bmatrix}$

in which the double underline indicates a second order tensor. This strain state is called plane strain. The corresponding stress tensor is:

$\underline{\underline{\sigma}} = \begin{bmatrix} \sigma_{11} & \sigma_{12} & 0 \\ \sigma_{21} & \sigma_{22} & 0 \\ 0 & 0 & \sigma_{33}\end{bmatrix}$

in which the non-zero $σ 33$ is needed to maintain the constraint $ε 33 = 0$ . This stress term can be temporarily removed from the analysis to leave only the in-plane terms, effectively reducing the 3-D problem to a much simpler 2-D problem.

[edit] See also

[edit] External links

v • d • e General subfields within physics

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Hidden categories: All pages needing to be wikified | Wikify from November 2007

Strain (materials science)

From Wikipedia, the free encyclopedia

Contents

[edit] Quantifying strain

[edit] Linear axial strain at single point

[edit] The general case of linear strain

[edit] Shear strain

[edit] Volumetric strain

[edit] The strain tensor

[edit] Principal strains in two dimensions

[edit] The case of large deformations

[edit] Compatibility equations

[edit] Engineering strain vs. true strain

[edit] Plane strain

[edit] See also

[edit] External links

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