Теория бесконечно малых деформаций

В механике сплошной среды , то теория бесконечно малой деформации представляет собой математический подход к описанию деформации твердого тела , в котором смещение материала частиц предполагаются значительно меньше ( на самом деле, бесконечно меньше) , чем любая соответствующая размерность тела; так что его геометрия и основные свойства материала (такие как плотность и жесткость ) в каждой точке пространства можно считать неизменными из-за деформации.

При таком предположении уравнения механики сплошной среды значительно упрощаются. Этот подход можно также назвать небольшую теорию деформации , теорию малых смещений , или небольшой теорию смещения-градиент . Это контрастирует с теорией конечных деформаций, в которой сделано противоположное предположение.

Теория бесконечно малых деформаций обычно применяется в гражданском строительстве и машиностроении для анализа напряжений конструкций, построенных из относительно жестких эластичных материалов, таких как бетон и сталь , поскольку общей целью при проектировании таких конструкций является минимизация их деформации при типичных нагрузках . Однако это приближение требует осторожности в случае тонких гибких тел, таких как стержни, пластины и оболочки, которые подвержены значительному вращению, что делает результаты ненадежными. ^[1]

Тензор бесконечно малых деформаций [ править ]

Для бесконечно малых деформаций одного тела континуума , в котором градиент смещения (тензор второго порядка) мал по сравнению с единицей, то есть , можно выполнить геометрическую линеаризацию любого одного из (бесконечного числа возможно) тензоров деформации , используемого в конечной деформации теории, например тензор лагранжевой деформации и тензор эйлеровой деформации . При такой линеаризации не учитываются нелинейные члены или члены второго порядка тензора конечных деформаций. Таким образом, мы имеем${\displaystyle \|\nabla \mathbf {u} \|\ll 1}$${\displaystyle \mathbf {E} }$${\displaystyle \mathbf {e} }$

${\displaystyle \mathbf {E} ={\frac {1}{2}}\left(\nabla _{\mathbf {X} }\mathbf {u} +(\nabla _{\mathbf {X} }\mathbf {u} )^{T}+(\nabla _{\mathbf {X} }\mathbf {u} )^{T}\nabla _{\mathbf {X} }\mathbf {u} \right)\approx {\frac {1}{2}}\left(\nabla _{\mathbf {X} }\mathbf {u} +(\nabla _{\mathbf {X} }\mathbf {u} )^{T}\right)}$

или же

${\displaystyle E_{KL}={\frac {1}{2}}\left({\frac {\partial U_{K}}{\partial X_{L}}}+{\frac {\partial U_{L}}{\partial X_{K}}}+{\frac {\partial U_{M}}{\partial X_{K}}}{\frac {\partial U_{M}}{\partial X_{L}}}\right)\approx {\frac {1}{2}}\left({\frac {\partial U_{K}}{\partial X_{L}}}+{\frac {\partial U_{L}}{\partial X_{K}}}\right)}$

а также

${\displaystyle \mathbf {e} ={\frac {1}{2}}\left(\nabla _{\mathbf {x} }\mathbf {u} +(\nabla _{\mathbf {x} }\mathbf {u} )^{T}-\nabla _{\mathbf {x} }\mathbf {u} (\nabla _{\mathbf {x} }\mathbf {u} )^{T}\right)\approx {\frac {1}{2}}\left(\nabla _{\mathbf {x} }\mathbf {u} +(\nabla _{\mathbf {x} }\mathbf {u} )^{T}\right)}$

или же

${\displaystyle e_{rs}={\frac {1}{2}}\left({\frac {\partial u_{r}}{\partial x_{s}}}+{\frac {\partial u_{s}}{\partial x_{r}}}-{\frac {\partial u_{k}}{\partial x_{r}}}{\frac {\partial u_{k}}{\partial x_{s}}}\right)\approx {\frac {1}{2}}\left({\frac {\partial u_{r}}{\partial x_{s}}}+{\frac {\partial u_{s}}{\partial x_{r}}}\right)}$

Эта линеаризация подразумевает, что лагранжевое описание и эйлерово описание примерно одинаковы, поскольку существует небольшая разница в материальных и пространственных координатах данной материальной точки в континууме. Следовательно, компоненты градиента смещения материала и компоненты градиента пространственного смещения приблизительно равны. Таким образом, мы имеем

${\displaystyle \mathbf {E} \approx \mathbf {e} \approx {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left((\nabla \mathbf {u} )^{T}+\nabla \mathbf {u} \right)\qquad }$

или же${\displaystyle \qquad E_{KL}\approx e_{rs}\approx \varepsilon _{ij}={\frac {1}{2}}\left(u_{i,j}+u_{j,i}\right)}$

где находятся компоненты тензора бесконечно малой деформации , также называемый тензор деформации Коши , линейный тензор деформации , или малого тензора деформации .${\displaystyle \varepsilon _{ij}}$ ${\displaystyle {\boldsymbol {\varepsilon }}}$

${\displaystyle {\begin{aligned}\varepsilon _{ij}&={\frac {1}{2}}\left(u_{i,j}+u_{j,i}\right)\\&=\left[{\begin{matrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\\\end{matrix}}\right]\\&=\left[{\begin{matrix}{\frac {\partial u_{1}}{\partial x_{1}}}&{\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{2}}}+{\frac {\partial u_{2}}{\partial x_{1}}}\right)&{\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{3}}}+{\frac {\partial u_{3}}{\partial x_{1}}}\right)\\{\frac {1}{2}}\left({\frac {\partial u_{2}}{\partial x_{1}}}+{\frac {\partial u_{1}}{\partial x_{2}}}\right)&{\frac {\partial u_{2}}{\partial x_{2}}}&{\frac {1}{2}}\left({\frac {\partial u_{2}}{\partial x_{3}}}+{\frac {\partial u_{3}}{\partial x_{2}}}\right)\\{\frac {1}{2}}\left({\frac {\partial u_{3}}{\partial x_{1}}}+{\frac {\partial u_{1}}{\partial x_{3}}}\right)&{\frac {1}{2}}\left({\frac {\partial u_{3}}{\partial x_{2}}}+{\frac {\partial u_{2}}{\partial x_{3}}}\right)&{\frac {\partial u_{3}}{\partial x_{3}}}\\\end{matrix}}\right]\end{aligned}}}$

или используя другие обозначения:

${\displaystyle \left[{\begin{matrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]=\left[{\begin{matrix}{\frac {\partial u_{x}}{\partial x}}&{\frac {1}{2}}\left({\frac {\partial u_{x}}{\partial y}}+{\frac {\partial u_{y}}{\partial x}}\right)&{\frac {1}{2}}\left({\frac {\partial u_{x}}{\partial z}}+{\frac {\partial u_{z}}{\partial x}}\right)\\{\frac {1}{2}}\left({\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}\right)&{\frac {\partial u_{y}}{\partial y}}&{\frac {1}{2}}\left({\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\right)\\{\frac {1}{2}}\left({\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}\right)&{\frac {1}{2}}\left({\frac {\partial u_{z}}{\partial y}}+{\frac {\partial u_{y}}{\partial z}}\right)&{\frac {\partial u_{z}}{\partial z}}\\\end{matrix}}\right]}$

Кроме того, поскольку градиент деформации может быть выражен как где - тождественный тензор второго порядка, мы имеем${\displaystyle {\boldsymbol {F}}={\boldsymbol {\nabla }}\mathbf {u} +{\boldsymbol {I}}}$${\displaystyle {\boldsymbol {I}}}$

${\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left({\boldsymbol {F}}^{T}+{\boldsymbol {F}}\right)-{\boldsymbol {I}}}$

Кроме того, из общего выражения для лагранжевых и эйлеровых тензоров конечных деформаций имеем

${\displaystyle {\begin{aligned}\mathbf {E} _{(m)}&={\frac {1}{2m}}(\mathbf {U} ^{2m}-{\boldsymbol {I}})={\frac {1}{2m}}[({\boldsymbol {F}}^{T}{\boldsymbol {F}})^{m}-{\boldsymbol {I}}]\approx {\frac {1}{2m}}[\{{\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{T}+{\boldsymbol {I}}\}^{m}-{\boldsymbol {I}}]\approx {\boldsymbol {\varepsilon }}\\\mathbf {e} _{(m)}&={\frac {1}{2m}}(\mathbf {V} ^{2m}-{\boldsymbol {I}})={\frac {1}{2m}}[({\boldsymbol {F}}{\boldsymbol {F}}^{T})^{m}-{\boldsymbol {I}}]\approx {\boldsymbol {\varepsilon }}\end{aligned}}}$

Геометрическое происхождение [ править ]

Рассмотрим двумерную деформацию бесконечно малого прямоугольного элемента материала с размерами от (фиг.1), который после деформации, принимает форму ромба. Из геометрии рисунка 1 мы имеем${\displaystyle dx}$${\displaystyle dy}$

${\displaystyle {\begin{aligned}{\overline {ab}}&={\sqrt {\left(dx+{\frac {\partial u_{x}}{\partial x}}dx\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}dx\right)^{2}}}\\&=dx{\sqrt {1+2{\frac {\partial u_{x}}{\partial x}}+\left({\frac {\partial u_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\\\end{aligned}}}$

Для очень малых градиентов смещения, то есть , мы имеем${\displaystyle \|\nabla \mathbf {u} \|\ll 1}$

${\displaystyle {\overline {ab}}\approx dx+{\frac {\partial u_{x}}{\partial x}}dx}$

Нормальное напряжение в направлении оси прямоугольного элемента определяются${\displaystyle x}$

${\displaystyle \varepsilon _{x}={\frac {{\overline {ab}}-{\overline {AB}}}{\overline {AB}}}}$

и зная это , у нас есть${\displaystyle {\overline {AB}}=dx}$

${\displaystyle \varepsilon _{x}={\frac {\partial u_{x}}{\partial x}}}$

Точно так же нормальная деформация в -направлении и -направлении становится${\displaystyle y}$${\displaystyle z}$

${\displaystyle \varepsilon _{y}={\frac {\partial u_{y}}{\partial y}}\quad ,\qquad \varepsilon _{z}={\frac {\partial u_{z}}{\partial z}}}$

В Engineering сдвиговой деформация , или изменение угла между двумя первоначально ортогональными материальными линиями, в этом случае линии и определяются как${\displaystyle {\overline {AC}}}$${\displaystyle {\overline {AB}}}$

${\displaystyle \gamma _{xy}=\alpha +\beta }$

Из геометрии рисунка 1 мы имеем

${\displaystyle \tan \alpha ={\frac {{\dfrac {\partial u_{y}}{\partial x}}dx}{dx+{\dfrac {\partial u_{x}}{\partial x}}dx}}={\frac {\dfrac {\partial u_{y}}{\partial x}}{1+{\dfrac {\partial u_{x}}{\partial x}}}}\quad ,\qquad \tan \beta ={\frac {{\dfrac {\partial u_{x}}{\partial y}}dy}{dy+{\dfrac {\partial u_{y}}{\partial y}}dy}}={\frac {\dfrac {\partial u_{x}}{\partial y}}{1+{\dfrac {\partial u_{y}}{\partial y}}}}}$

Для малых поворотов, то есть и есть мы имеем${\displaystyle \alpha }$${\displaystyle \beta }$${\displaystyle \ll 1}$

${\displaystyle \tan \alpha \approx \alpha \quad ,\qquad \tan \beta \approx \beta }$

и, опять же, для малых градиентов смещения имеем

${\displaystyle \alpha ={\frac {\partial u_{y}}{\partial x}}\quad ,\qquad \beta ={\frac {\partial u_{x}}{\partial y}}}$

таким образом

${\displaystyle \gamma _{xy}=\alpha +\beta ={\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}}$

Меняя местами и и и , можно показать, что${\displaystyle x}$${\displaystyle y}$${\displaystyle u_{x}}$${\displaystyle u_{y}}$${\displaystyle \gamma _{xy}=\gamma _{yx}}$

Точно так же для самолетов - и - у нас есть${\displaystyle y}$${\displaystyle z}$${\displaystyle x}$${\displaystyle z}$

${\displaystyle \gamma _{yz}=\gamma _{zy}={\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\quad ,\qquad \gamma _{zx}=\gamma _{xz}={\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}}$

Можно видеть, что компоненты тензорной деформации сдвига тензора бесконечно малых деформаций могут быть затем выражены с использованием определения инженерной деформации , как${\displaystyle \gamma }$

${\displaystyle \left[{\begin{matrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]=\left[{\begin{matrix}\varepsilon _{xx}&\gamma _{xy}/2&\gamma _{xz}/2\\\gamma _{yx}/2&\varepsilon _{yy}&\gamma _{yz}/2\\\gamma _{zx}/2&\gamma _{zy}/2&\varepsilon _{zz}\\\end{matrix}}\right]}$

Физическая интерпретация [ править ]

Из теории конечных деформаций имеем

${\displaystyle d\mathbf {x} ^{2}-d\mathbf {X} ^{2}=d\mathbf {X} \cdot 2\mathbf {E} \cdot d\mathbf {X} \quad {\text{or}}\quad (dx)^{2}-(dX)^{2}=2E_{KL}\,dX_{K}\,dX_{L}}$

Для бесконечно малых деформаций имеем

${\displaystyle d\mathbf {x} ^{2}-d\mathbf {X} ^{2}=d\mathbf {X} \cdot 2\mathbf {\boldsymbol {\varepsilon }} \cdot d\mathbf {X} \quad {\text{or}}\quad (dx)^{2}-(dX)^{2}=2\varepsilon _{KL}\,dX_{K}\,dX_{L}}$

Разделив на, мы имеем${\displaystyle (dX)^{2}}$

${\displaystyle {\frac {dx-dX}{dX}}{\frac {dx+dX}{dX}}=2\varepsilon _{ij}{\frac {dX_{i}}{dX}}{\frac {dX_{j}}{dX}}}$

Для малых деформаций мы предполагаем , что , таким образом, второй член с левой стороны будет выглядеть так : .${\displaystyle dx\approx dX}$${\displaystyle {\frac {dx+dX}{dX}}\approx 2}$

Тогда у нас есть

${\displaystyle {\frac {dx-dX}{dX}}=\varepsilon _{ij}N_{i}N_{j}=\mathbf {N} \cdot {\boldsymbol {\varepsilon }}\cdot \mathbf {N} }$

где , - единичный вектор в направлении , а выражение в левой части - нормальная деформация в направлении . Для частного случая в направлении, т. Е. Имеем${\displaystyle N_{i}={\frac {dX_{i}}{dX}}}$${\displaystyle d\mathbf {X} }$ ${\displaystyle e_{(\mathbf {N} )}}$${\displaystyle \mathbf {N} }$${\displaystyle \mathbf {N} }$${\displaystyle X_{1}}$${\displaystyle \mathbf {N} =\mathbf {I} _{1}}$

${\displaystyle e_{(\mathbf {I} _{1})}=\mathbf {I} _{1}\cdot {\boldsymbol {\varepsilon }}\cdot \mathbf {I} _{1}=\varepsilon _{11}}$

Аналогично для и мы можем найти нормальные деформации и , соответственно. Следовательно, диагональные элементы тензора бесконечно малых деформаций являются нормальными деформациями в координатных направлениях.${\displaystyle \mathbf {N} =\mathbf {I} _{2}}$${\displaystyle \mathbf {N} =\mathbf {I} _{3}}$${\displaystyle \varepsilon _{22}}$${\displaystyle \varepsilon _{33}}$

Правила трансформации штамма [ править ]

Если мы выберем ортонормированную систему координат ( ), мы можем записать тензор в терминах компонентов по отношению к этим базовым векторам как${\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}}$

${\displaystyle {\boldsymbol {\varepsilon }}=\sum _{i=1}^{3}\sum _{j=1}^{3}\varepsilon _{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}}$

В матричной форме

${\displaystyle {\underline {\underline {\boldsymbol {\varepsilon }}}}={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{12}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{13}&\varepsilon _{23}&\varepsilon _{33}\end{bmatrix}}}$

Вместо этого мы можем легко выбрать другую ортонормированную систему координат ( ). В этом случае компоненты тензора разные, скажем${\displaystyle {\hat {\mathbf {e} }}_{1},{\hat {\mathbf {e} }}_{2},{\hat {\mathbf {e} }}_{3}}$

${\displaystyle {\boldsymbol {\varepsilon }}=\sum _{i=1}^{3}\sum _{j=1}^{3}{\hat {\varepsilon }}_{ij}{\hat {\mathbf {e} }}_{i}\otimes {\hat {\mathbf {e} }}_{j}\quad \implies \quad {\underline {\underline {\hat {\boldsymbol {\varepsilon }}}}}={\begin{bmatrix}{\hat {\varepsilon }}_{11}&{\hat {\varepsilon }}_{12}&{\hat {\varepsilon }}_{13}\\{\hat {\varepsilon }}_{12}&{\hat {\varepsilon }}_{22}&{\hat {\varepsilon }}_{23}\\{\hat {\varepsilon }}_{13}&{\hat {\varepsilon }}_{23}&{\hat {\varepsilon }}_{33}\end{bmatrix}}}$

Компоненты деформации в двух системах координат связаны соотношением

${\displaystyle {\hat {\varepsilon }}_{ij}=\ell _{ip}~\ell _{jq}~\varepsilon _{pq}}$

where the Einstein summation convention for repeated indices has been used and ${\displaystyle \ell _{ij}={\hat {\mathbf {e} }}_{i}\cdot \mathbf {e} _{j}}$. In matrix form

${\displaystyle {\underline {\underline {\hat {\boldsymbol {\varepsilon }}}}}={\underline {\underline {\mathbf {L} }}}~{\underline {\underline {\boldsymbol {\varepsilon }}}}~{\underline {\underline {\mathbf {L} }}}^{T}}$

or

${\displaystyle {\begin{bmatrix}{\hat {\varepsilon }}_{11}&{\hat {\varepsilon }}_{12}&{\hat {\varepsilon }}_{13}\\{\hat {\varepsilon }}_{21}&{\hat {\varepsilon }}_{22}&{\hat {\varepsilon }}_{23}\\{\hat {\varepsilon }}_{31}&{\hat {\varepsilon }}_{32}&{\hat {\varepsilon }}_{33}\end{bmatrix}}={\begin{bmatrix}\ell _{11}&\ell _{12}&\ell _{13}\\\ell _{21}&\ell _{22}&\ell _{23}\\\ell _{31}&\ell _{32}&\ell _{33}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}{\begin{bmatrix}\ell _{11}&\ell _{12}&\ell _{13}\\\ell _{21}&\ell _{22}&\ell _{23}\\\ell _{31}&\ell _{32}&\ell _{33}\end{bmatrix}}^{T}}$

Strain invariants[edit]

Certain operations on the strain tensor give the same result without regard to which orthonormal coordinate system is used to represent the components of strain. The results of these operations are called strain invariants. The most commonly used strain invariants are

${\displaystyle {\begin{aligned}I_{1}&=\mathrm {tr} ({\boldsymbol {\varepsilon }})\\I_{2}&={\tfrac {1}{2}}\{[\mathrm {tr} ({\boldsymbol {\varepsilon }})]^{2}-\mathrm {tr} ({\boldsymbol {\varepsilon }}^{2})\}\\I_{3}&=\det({\boldsymbol {\varepsilon }})\end{aligned}}}$

In terms of components

${\displaystyle {\begin{aligned}I_{1}&=\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}\\I_{2}&=\varepsilon _{12}^{2}+\varepsilon _{23}^{2}+\varepsilon _{31}^{2}-\varepsilon _{11}\varepsilon _{22}-\varepsilon _{22}\varepsilon _{33}-\varepsilon _{33}\varepsilon _{11}\\I_{3}&=\varepsilon _{11}(\varepsilon _{22}\varepsilon _{33}-\varepsilon _{23}^{2})-\varepsilon _{12}(\varepsilon _{11}\varepsilon _{33}-\varepsilon _{23}\varepsilon _{31})+\varepsilon _{13}(\varepsilon _{21}\varepsilon _{32}-\varepsilon _{22}\varepsilon _{31})\end{aligned}}}$

Principal strains[edit]

It can be shown that it is possible to find a coordinate system (${\displaystyle \mathbf {n} _{1},\mathbf {n} _{2},\mathbf {n} _{3}}$) in which the components of the strain tensor are

${\displaystyle {\underline {\underline {\boldsymbol {\varepsilon }}}}={\begin{bmatrix}\varepsilon _{1}&0&0\\0&\varepsilon _{2}&0\\0&0&\varepsilon _{3}\end{bmatrix}}\quad \implies \quad {\boldsymbol {\varepsilon }}=\varepsilon _{1}\mathbf {n} _{1}\otimes \mathbf {n} _{1}+\varepsilon _{2}\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\varepsilon _{3}\mathbf {n} _{3}\otimes \mathbf {n} _{3}}$

The components of the strain tensor in the (${\displaystyle \mathbf {n} _{1},\mathbf {n} _{2},\mathbf {n} _{3}}$) coordinate system are called the principal strains and the directions ${\displaystyle \mathbf {n} _{i}}$ are called the directions of principal strain. Since there are no shear strain components in this coordinate system, the principal strains represent the maximum and minimum stretches of an elemental volume.

If we are given the components of the strain tensor in an arbitrary orthonormal coordinate system, we can find the principal strains using an eigenvalue decomposition determined by solving the system of equations

${\displaystyle ({\underline {\underline {\boldsymbol {\varepsilon }}}}-\varepsilon _{i}~{\underline {\underline {\mathbf {I} }}})~\mathbf {n} _{i}={\underline {\mathbf {0} }}}$

This system of equations is equivalent to finding the vector ${\displaystyle \mathbf {n} _{i}}$ along which the strain tensor becomes a pure stretch with no shear component.

Volumetric strain[edit]

The dilatation (the relative variation of the volume) is the first strain invariant or trace of the tensor:

${\displaystyle \delta ={\frac {\Delta V}{V_{0}}}=I_{1}=\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}}$

Actually, if we consider a cube with an edge length a, it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions ${\displaystyle a\cdot (1+\varepsilon _{11})\times a\cdot (1+\varepsilon _{22})\times a\cdot (1+\varepsilon _{33})}$ and V₀ = a³, thus

${\displaystyle {\frac {\Delta V}{V_{0}}}={\frac {\left(1+\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}+\varepsilon _{11}\cdot \varepsilon _{33}+\varepsilon _{22}\cdot \varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}\right)\cdot a^{3}-a^{3}}{a^{3}}}}$

as we consider small deformations,

${\displaystyle 1\gg \varepsilon _{ii}\gg \varepsilon _{ii}\cdot \varepsilon _{jj}\gg \varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}}$

therefore the formula.

Real variation of volume (top) and the approximated one (bottom): the green drawing shows the estimated volume and the orange drawing the neglected volume

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

Strain deviator tensor[edit]

The infinitesimal strain tensor ${\displaystyle \varepsilon _{ij}}$, similarly to the Cauchy stress tensor, can be expressed as the sum of two other tensors:

1. a mean strain tensor or volumetric strain tensor or spherical strain tensor, ${\displaystyle \varepsilon _{M}\delta _{ij}}$, related to dilation or volume change; and
2. a deviatoric component called the strain deviator tensor, ${\displaystyle \varepsilon '_{ij}}$, related to distortion.

${\displaystyle \varepsilon _{ij}=\varepsilon '_{ij}+\varepsilon _{M}\delta _{ij}}$

where ${\displaystyle \varepsilon _{M}}$ is the mean strain given by

${\displaystyle \varepsilon _{M}={\frac {\varepsilon _{kk}}{3}}={\frac {\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}}{3}}={\tfrac {1}{3}}I_{1}^{e}}$

The deviatoric strain tensor can be obtained by subtracting the mean strain tensor from the infinitesimal strain tensor:

${\displaystyle {\begin{aligned}\ \varepsilon '_{ij}&=\varepsilon _{ij}-{\frac {\varepsilon _{kk}}{3}}\delta _{ij}\\\left[{\begin{matrix}\varepsilon '_{11}&\varepsilon '_{12}&\varepsilon '_{13}\\\varepsilon '_{21}&\varepsilon '_{22}&\varepsilon '_{23}\\\varepsilon '_{31}&\varepsilon '_{32}&\varepsilon '_{33}\\\end{matrix}}\right]&=\left[{\begin{matrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\\\end{matrix}}\right]-\left[{\begin{matrix}\varepsilon _{M}&0&0\\0&\varepsilon _{M}&0\\0&0&\varepsilon _{M}\\\end{matrix}}\right]\\&=\left[{\begin{matrix}\varepsilon _{11}-\varepsilon _{M}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}-\varepsilon _{M}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}-\varepsilon _{M}\\\end{matrix}}\right]\\\end{aligned}}}$

Octahedral strains[edit]

Let (${\displaystyle \mathbf {n} _{1},\mathbf {n} _{2},\mathbf {n} _{3}}$) be the directions of the three principal strains. An octahedral plane is one whose normal makes equal angles with the three principal directions. The engineering shear strain on an octahedral plane is called the octahedral shear strain and is given by

${\displaystyle \gamma _{\mathrm {oct} }={\tfrac {2}{3}}{\sqrt {(\varepsilon _{1}-\varepsilon _{2})^{2}+(\varepsilon _{2}-\varepsilon _{3})^{2}+(\varepsilon _{3}-\varepsilon _{1})^{2}}}}$

where ${\displaystyle \varepsilon _{1},\varepsilon _{2},\varepsilon _{3}}$ are the principal strains.^{[citation needed]}

The normal strain on an octahedral plane is given by

${\displaystyle \varepsilon _{\mathrm {oct} }={\tfrac {1}{3}}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})}$^{[citation needed]}

Equivalent strain[edit]

A scalar quantity called the equivalent strain, or the von Mises equivalent strain, is often used to describe the state of strain in solids. Several definitions of equivalent strain can be found in the literature. A definition that is commonly used in the literature on plasticity is

${\displaystyle \varepsilon _{\mathrm {eq} }={\sqrt {{\tfrac {2}{3}}{\boldsymbol {\varepsilon }}^{\mathrm {dev} }:{\boldsymbol {\varepsilon }}^{\mathrm {dev} }}}={\sqrt {{\tfrac {2}{3}}\varepsilon _{ij}^{\mathrm {dev} }\varepsilon _{ij}^{\mathrm {dev} }}}~;~~{\boldsymbol {\varepsilon }}^{\mathrm {dev} }={\boldsymbol {\varepsilon }}-{\tfrac {1}{3}}\mathrm {tr} ({\boldsymbol {\varepsilon }})~{\boldsymbol {I}}}$

This quantity is work conjugate to the equivalent stress defined as

${\displaystyle \sigma _{\mathrm {eq} }={\sqrt {{\tfrac {3}{2}}{\boldsymbol {\sigma }}^{\mathrm {dev} }:{\boldsymbol {\sigma }}^{\mathrm {dev} }}}}$

Compatibility equations[edit]

For prescribed strain components ${\displaystyle \varepsilon _{ij}}$ the strain tensor equation ${\displaystyle u_{i,j}+u_{j,i}=2\varepsilon _{ij}}$ represents a system of six differential equations for the determination of three displacements components ${\displaystyle u_{i}}$, 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 are 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 ${\displaystyle u_{i}}$. If the elastic medium is visualised 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

${\displaystyle \varepsilon _{ij,km}+\varepsilon _{km,ij}-\varepsilon _{ik,jm}-\varepsilon _{jm,ik}=0}$

Engineering notation

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

Special cases[edit]

Plane strain[edit]

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 ${\displaystyle \varepsilon _{33}}$ and the shear strains ${\displaystyle \varepsilon _{13}}$ and ${\displaystyle \varepsilon _{23}}$ (if the length is the 3-direction) are constrained by nearby material and are small compared to the cross-sectional strains. Plane strain is then an acceptable approximation. The strain tensor for plane strain is written as:

${\displaystyle {\underline {\underline {\boldsymbol {\varepsilon }}}}={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&0\\\varepsilon _{21}&\varepsilon _{22}&0\\0&0&0\end{bmatrix}}}$

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

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

in which the non-zero ${\displaystyle \sigma _{33}}$ is needed to maintain the constraint ${\displaystyle \epsilon _{33}=0}$. 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.

Antiplane strain[edit]

Antiplane strain is another special state of strain that can occur in a body, for instance in a region close to a screw dislocation. The strain tensor for antiplane strain is given by

${\displaystyle {\underline {\underline {\boldsymbol {\varepsilon }}}}={\begin{bmatrix}0&0&\varepsilon _{13}\\0&0&\varepsilon _{23}\\\varepsilon _{13}&\varepsilon _{23}&0\end{bmatrix}}}$

Infinitesimal rotation tensor[edit]

The infinitesimal strain tensor is defined as

${\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2}}[{\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{T}]}$

Therefore the displacement gradient can be expressed as

${\displaystyle {\boldsymbol {\nabla }}\mathbf {u} ={\boldsymbol {\varepsilon }}+{\boldsymbol {\omega }}}$

where

${\displaystyle {\boldsymbol {\omega }}:={\frac {1}{2}}[{\boldsymbol {\nabla }}\mathbf {u} -({\boldsymbol {\nabla }}\mathbf {u} )^{T}]}$

The quantity ${\displaystyle {\boldsymbol {\omega }}}$ is the infinitesimal rotation tensor. This tensor is skew symmetric. For infinitesimal deformations the scalar components of ${\displaystyle {\boldsymbol {\omega }}}$ satisfy the condition ${\displaystyle |\omega _{ij}|\ll 1}$. Note that the displacement gradient is small only if both the strain tensor and the rotation tensor are infinitesimal.

The axial vector[edit]

A skew symmetric second-order tensor has three independent scalar components. These three components are used to define an axial vector, ${\displaystyle \mathbf {w} }$, as follows

${\displaystyle \omega _{ij}=-\epsilon _{ijk}~w_{k}~;~~w_{i}=-{\tfrac {1}{2}}~\epsilon _{ijk}~\omega _{jk}}$

where ${\displaystyle \epsilon _{ijk}}$ is the permutation symbol. In matrix form

${\displaystyle {\underline {\underline {\boldsymbol {\omega }}}}={\begin{bmatrix}0&-w_{3}&w_{2}\\w_{3}&0&-w_{1}\\-w_{2}&w_{1}&0\end{bmatrix}}~;~~{\underline {\mathbf {w} }}={\begin{bmatrix}w_{1}\\w_{2}\\w_{3}\end{bmatrix}}}$

The axial vector is also called the infinitesimal rotation vector. The rotation vector is related to the displacement gradient by the relation

${\displaystyle \mathbf {w} ={\tfrac {1}{2}}~{\boldsymbol {\nabla }}\times \mathbf {u} }$

In index notation

${\displaystyle w_{i}={\tfrac {1}{2}}~\epsilon _{ijk}~u_{k,j}}$

If ${\displaystyle \lVert {\boldsymbol {\omega }}\rVert \ll 1}$ and ${\displaystyle {\boldsymbol {\varepsilon }}={\boldsymbol {0}}}$ then the material undergoes an approximate rigid body rotation of magnitude ${\displaystyle |\mathbf {w} |}$ around the vector ${\displaystyle \mathbf {w} }$.

Relation between the strain tensor and the rotation vector[edit]

Given a continuous, single-valued displacement field ${\displaystyle \mathbf {u} }$ and the corresponding infinitesimal strain tensor ${\displaystyle {\boldsymbol {\varepsilon }}}$, we have (see Tensor derivative (continuum mechanics))

${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {\varepsilon }}=e_{ijk}~\varepsilon _{lj,i}~\mathbf {e} _{k}\otimes \mathbf {e} _{l}={\tfrac {1}{2}}~e_{ijk}~[u_{l,ji}+u_{j,li}]~\mathbf {e} _{k}\otimes \mathbf {e} _{l}}$

Since a change in the order of differentiation does not change the result, ${\displaystyle u_{l,ji}=u_{l,ij}}$. Therefore

${\displaystyle \,e_{ijk}u_{l,ji}=(e_{12k}+e_{21k})u_{l,12}+(e_{13k}+e_{31k})u_{l,13}+(e_{23k}+e_{32k})u_{l,32}=0}$

Also

${\displaystyle {\tfrac {1}{2}}~e_{ijk}~u_{j,li}=\left({\tfrac {1}{2}}~e_{ijk}~u_{j,i}\right)_{,l}=\left({\tfrac {1}{2}}~e_{kij}~u_{j,i}\right)_{,l}=w_{k,l}}$

Hence

${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {\varepsilon }}=w_{k,l}~\mathbf {e} _{k}\otimes \mathbf {e} _{l}={\boldsymbol {\nabla }}\mathbf {w} }$

Relation between rotation tensor and rotation vector[edit]

From an important identity regarding the curl of a tensor we know that for a continuous, single-valued displacement field ${\displaystyle \mathbf {u} }$,

${\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\mathbf {u} )={\boldsymbol {0}}.}$

Since ${\displaystyle {\boldsymbol {\nabla }}\mathbf {u} ={\boldsymbol {\varepsilon }}+{\boldsymbol {\omega }}}$ we have${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {\omega }}=-{\boldsymbol {\nabla }}\times {\boldsymbol {\varepsilon }}=-{\boldsymbol {\nabla }}\mathbf {w} .}$

Strain tensor in cylindrical coordinates[edit]

In cylindrical polar coordinates (${\displaystyle r,\theta ,z}$), the displacement vector can be written as

${\displaystyle \mathbf {u} =u_{r}~\mathbf {e} _{r}+u_{\theta }~\mathbf {e} _{\theta }+u_{z}~\mathbf {e} _{z}}$

The components of the strain tensor in a cylindrical coordinate system are given by:^[2]${\displaystyle {\begin{aligned}\varepsilon _{rr}&={\cfrac {\partial u_{r}}{\partial r}}\\\varepsilon _{\theta \theta }&={\cfrac {1}{r}}\left({\cfrac {\partial u_{\theta }}{\partial \theta }}+u_{r}\right)\\\varepsilon _{zz}&={\cfrac {\partial u_{z}}{\partial z}}\\\varepsilon _{r\theta }&={\cfrac {1}{2}}\left({\cfrac {1}{r}}{\cfrac {\partial u_{r}}{\partial \theta }}+{\cfrac {\partial u_{\theta }}{\partial r}}-{\cfrac {u_{\theta }}{r}}\right)\\\varepsilon _{\theta z}&={\cfrac {1}{2}}\left({\cfrac {\partial u_{\theta }}{\partial z}}+{\cfrac {1}{r}}{\cfrac {\partial u_{z}}{\partial \theta }}\right)\\\varepsilon _{zr}&={\cfrac {1}{2}}\left({\cfrac {\partial u_{r}}{\partial z}}+{\cfrac {\partial u_{z}}{\partial r}}\right)\end{aligned}}}$

Strain tensor in spherical coordinates[edit]

In spherical coordinates (${\displaystyle r,\theta ,\phi }$), the displacement vector can be written as

${\displaystyle \mathbf {u} =u_{r}~\mathbf {e} _{r}+u_{\theta }~\mathbf {e} _{\theta }+u_{\phi }~\mathbf {e} _{\phi }}$

The components of the strain tensor in a spherical coordinate system are given by ^[2]

${\displaystyle {\begin{aligned}\varepsilon _{rr}&={\cfrac {\partial u_{r}}{\partial r}}\\\varepsilon _{\theta \theta }&={\cfrac {1}{r}}\left({\cfrac {\partial u_{\theta }}{\partial \theta }}+u_{r}\right)\\\varepsilon _{\phi \phi }&={\cfrac {1}{r\sin \theta }}\left({\cfrac {\partial u_{\phi }}{\partial \phi }}+u_{r}\sin \theta +u_{\theta }\cos \theta \right)\\\varepsilon _{r\theta }&={\cfrac {1}{2}}\left({\cfrac {1}{r}}{\cfrac {\partial u_{r}}{\partial \theta }}+{\cfrac {\partial u_{\theta }}{\partial r}}-{\cfrac {u_{\theta }}{r}}\right)\\\varepsilon _{\theta \phi }&={\cfrac {1}{2r}}\left({\cfrac {1}{\sin \theta }}{\cfrac {\partial u_{\theta }}{\partial \phi }}+{\cfrac {\partial u_{\phi }}{\partial \theta }}-u_{\phi }\cot \theta \right)\\\varepsilon _{\phi r}&={\cfrac {1}{2}}\left({\cfrac {1}{r\sin \theta }}{\cfrac {\partial u_{r}}{\partial \phi }}+{\cfrac {\partial u_{\phi }}{\partial r}}-{\cfrac {u_{\phi }}{r}}\right)\end{aligned}}}$