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Strain (mechanics)

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Relative deformation of a physical body

Strain
Other names	Strain tensor
SI unit	1
Other units	%
InSI base units	m/m
Behaviour under coord transformation	tensor
Dimension	$1 {\displaystyle 1}$

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Inmechanics,strain is defined as relativedeformation, compared to areferenceposition configuration. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether themetric tensor or its dual is considered.

Strain hasdimension of alength ratio, withSI base units of meter per meter (m/m).Hence strains aredimensionless and are usually expressed as adecimal fraction or apercentage.Parts-per notation is also used, e.g.,parts per million orparts per billion (sometimes called "microstrains" and "nanostrains", respectively), corresponding toμm/m andnm/m.

Strain can be formulated as thespatial derivative ofdisplacement: ${\boldsymbol {\varepsilon }}\doteq {\cfrac {\partial }{\partial \mathbf {X} }}\left(\mathbf {x} -\mathbf {X} \right)={\boldsymbol {F}}'-{\boldsymbol {I}},$ whereI is theidentity tensor.The displacement of a body may be expressed in the formx =F(X), whereX is the reference position of material points of the body; displacement has units of length and does not distinguish between rigid body motions (translations and rotations) and deformations (changes in shape and size) of the body.The spatial derivative of a uniform translation is zero, thus strains measure how much a given displacement differs locally from a rigid-body motion.^[1]

A strain is in general atensor quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along material line elements or fibers is thenormal strain, and the amount of distortion associated with the sliding of plane layers over each other is theshear strain, within a deforming body.^[2] This could be applied by elongation, shortening, or volume changes, or angular distortion.^[3]

The state of strain at amaterial point of a continuum body is defined as the totality of all the changes in length of material lines or fibers, thenormal strain, which pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other, theshear strain, radiating from this point. However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions.

If there is an increase in length of the material line, the normal strain is calledtensile strain; otherwise, if there is reduction or compression in the length of the material line, it is calledcompressive strain.

Strain regimes

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Depending on the amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories:

Finite strain theory, also calledlarge strain theory,large deformation theory, deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and deformed configurations of thecontinuum are significantly different and a clear distinction has to be made between them. This is commonly the case withelastomers,plastically-deforming materials and otherfluids and biologicalsoft tissue.
Infinitesimal strain theory, also calledsmall strain theory,small deformation theory,small displacement theory, orsmall displacement-gradient theory where strains and rotations are both small. In this case, the undeformed and deformed configurations of the body can be assumed identical. The infinitesimal strain theory is used in the analysis of deformations of materials exhibitingelastic behavior, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel.
Large-displacement orlarge-rotation theory, which assumes small strains but large rotations and displacements.

Strain measures

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In each of these theories the strain is then defined differently. Theengineering strain is the most common definition applied to materials used in mechanical and structural engineering, which are subjected to very small deformations. On the other hand, for some materials, e.g.,elastomers and polymers, subjected to large deformations, the engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%;^[4] thus other more complex definitions of strain are required, such asstretch,logarithmic strain,Green strain, andAlmansi strain.

Engineering strain

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Engineering strain, also known asCauchy strain, is expressed as the ratio of total deformation to the initial dimension of the material body on which forces are applied. In the case of a material line element or fiber axially loaded, itselongation gives rise to anengineering normal strain orengineering extensional straine, which equals therelative elongation or the change in lengthΔL per unit of the original lengthL of the line element or fibers (in meters per meter). The normal strain is positive if the material fibers are stretched and negative if they are compressed. Thus, we have $e={\frac {\Delta L}{L}}={\frac {l-L}{L}}$ ,wheree is theengineering normal strain,L is the original length of the fiber andl is the final length of the fiber.

Thetrue shear strain is defined as the change in the angle (in radians) between two material line elements initially perpendicular to each other in the undeformed or initial configuration. Theengineering shear strain is defined as the tangent of that angle, and is equal to the length of deformation at its maximum divided by the perpendicular length in the plane of force application, which sometimes makes it easier to calculate.

Stretch ratio

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Thestretch ratio orextension ratio (symbol λ) is an alternative measure related to the extensional or normal strain of an axially loaded differential line element. It is defined as the ratio between the final lengthl and the initial lengthL of the material line. $\lambda ={\frac {l}{L}}$

The extension ratio λ is related to the engineering straine by $e=\lambda -1$ This equation implies that when the normal strain is zero, so that there is no deformation, the stretch ratio is equal to unity.

The stretch ratio is used in the analysis of materials that exhibit large deformations, such aselastomers, which can sustain stretch ratios of 3 or 4 before they fail. On the other hand, traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios.

Logarithmic strain

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Thelogarithmic strainε, also called,true strain orHencky strain.^[5] Considering an incremental strain (Ludwik) $\delta \varepsilon ={\frac {\delta l}{l}}$ the logarithmic strain is obtained by integrating this incremental strain: ${\begin{aligned}\int \delta \varepsilon &=\int _{L}^{l}{\frac {\delta l}{l}}\\\varepsilon &=\ln \left({\frac {l}{L}}\right)=\ln(\lambda )\\&=\ln(1+e)\\&=e-{\frac {e^{2}}{2}}+{\frac {e^{3}}{3}}-\cdots \end{aligned}}$ wheree is the engineering strain. The logarithmic strain provides the correct measure of the final strain when deformation takes place in a series of increments, taking into account the influence of the strain path.^[2]

Green strain

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Main article:Finite strain theory

The Green strain is defined as: $\varepsilon _{G}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{L^{2}}}\right)={\tfrac {1}{2}}(\lambda ^{2}-1)$

Almansi strain

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Main article:Finite strain theory

The Euler-Almansi strain is defined as $\varepsilon _{E}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{l^{2}}}\right)={\tfrac {1}{2}}\left(1-{\frac {1}{\lambda ^{2}}}\right)$

Strain tensor

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Further information:Infinitesimal strain theory § Infinitesimal strain tensor

The (infinitesimal)strain tensor (symbol ${\boldsymbol {\varepsilon }}$ ) is defined in theInternational System of Quantities (ISQ), more specifically inISO 80000-4 (Mechanics), as a "tensor quantity representing the deformation of matter caused by stress. Strain tensor is symmetric and has three linear strain and three shear strain (Cartesian) components."^[6]ISO 80000-4 further defineslinear strain as the "quotient of change in length of an object and its length" andshear strain as the "quotient of parallel displacement of two surfaces of a layer and the thickness of the layer".^[6]Thus, strains are classified as eithernormal orshear. Anormal strain is perpendicular to the face of an element, and ashear strain is parallel to it. These definitions are consistent with those ofnormal stress andshear stress.

The strain tensor can then be expressed in terms of normal and shear components as: ${\underline {\underline {\boldsymbol {\varepsilon }}}}={\begin{bmatrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{bmatrix}}={\begin{bmatrix}\varepsilon _{xx}&{\tfrac {1}{2}}\gamma _{xy}&{\tfrac {1}{2}}\gamma _{xz}\\{\tfrac {1}{2}}\gamma _{yx}&\varepsilon _{yy}&{\tfrac {1}{2}}\gamma _{yz}\\{\tfrac {1}{2}}\gamma _{zx}&{\tfrac {1}{2}}\gamma _{zy}&\varepsilon _{zz}\\\end{bmatrix}}$

Geometric setting

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Two-dimensional geometric deformation of an infinitesimal material element

Consider a two-dimensional, infinitesimal, rectangular material element with dimensionsdx ×dy, which, after deformation, takes the form of arhombus. The deformation is described by thedisplacement fieldu. From the geometry of the adjacent figure we have $\mathrm {length} (AB)=dx$ and ${\begin{aligned}\mathrm {length} (ab)&={\sqrt {\left(dx+{\frac {\partial u_{x}}{\partial x}}dx\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}dx\right)^{2}}}\\&={\sqrt {dx^{2}\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+dx^{2}\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\\&=dx~{\sqrt {\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\end{aligned}}$ For very small displacement gradients the squares of the derivative of $u_{y}$ and $u_{x}$ are negligible and we have $\mathrm {length} (ab)\approx dx\left(1+{\frac {\partial u_{x}}{\partial x}}\right)=dx+{\frac {\partial u_{x}}{\partial x}}dx$

Normal strain

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For anisotropic material that obeysHooke's law, anormal stress will cause a normal strain. Normal strains producedilations.

The normal strain in thex-direction of the rectangular element is defined by $\varepsilon _{x}={\frac {\text{extension}}{\text{original length}}}={\frac {\mathrm {length} (ab)-\mathrm {length} (AB)}{\mathrm {length} (AB)}}={\frac {\partial u_{x}}{\partial x}}$ Similarly, the normal strain in they- andz-directions becomes $\varepsilon _{y}={\frac {\partial u_{y}}{\partial y}}\quad ,\qquad \varepsilon _{z}={\frac {\partial u_{z}}{\partial z}}$

Shear strain

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Shear strain
Common symbols	γ orε
SI unit	1, orradian
Derivations from other quantities	γ =⁠τ/G⁠

Volume strain

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This section is an excerpt fromInfinitesimal strain theory § Volumetric strain.[edit]

The volumetric strain, also called bulk strain, is the relative variation of the volume, as arising fromdilation orcompression; it is thefirst strain invariant ortrace of the tensor: $\delta ={\frac {\Delta V}{V_{0}}}=I_{1}=\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}$ Actually, if we consider a cube with an edge lengtha, it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions $a\cdot (1+\varepsilon _{11})\times a\cdot (1+\varepsilon _{22})\times a\cdot (1+\varepsilon _{33})$ andV₀ =a³, thus ${\frac {\Delta V}{V_{0}}}={\frac {\left(1+\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}+\varepsilon _{11}\cdot \varepsilon _{33}+\varepsilon _{22}\cdot \varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}\right)\cdot a^{3}-a^{3}}{a^{3}}}$ as we consider small deformations, $1\gg \varepsilon _{ii}\gg \varepsilon _{ii}\cdot \varepsilon _{jj}\gg \varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}$ therefore the formula.

In case of pure shear, we can see that there is no change of the volume.

Metric tensor

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Main article:Finite strain theory § Deformation tensors in curvilinear coordinates

A strain field associated with a displacement is defined, at any point, by the change in length of thetangent vectors representing the speeds of arbitrarilyparametrized curves passing through that point. A basic geometric result, due toFréchet,von Neumann andJordan, states that, if the lengths of the tangent vectors fulfil the axioms of anorm and theparallelogram law, then the length of a vector is the square root of the value of thequadratic form associated, by thepolarization formula, with apositive definite bilinear map called themetric tensor.