In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness) at each point of space can be assumed to be unchanged by the deformation. With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called small deformation theory, small displacement theory, or small displacement-gradient theory. It is contrasted with the finite strain theory where the opposite assumption is made. The infinitesimal strain theory has wide applications in engineering. Stress analysis, for example, tries to predict the behavior of structures built from relatively stiff elastic materials, such as concrete and steel. Such analysis can be used to minimize a structure design's deformation under typical loads. However, this approximation demands caution in the case of thin flexible bodies, such as rods, plates, and shells which are susceptible to significant rotations, thus making the results unreliable.
Infinitesimal strain tensor For infinitesimal deformations of a continuum body, in which the displacement gradient tensor (2nd order tensor) is small compared to unity, i.e. ‖ ∇ u ‖ ≪ 1 {\displaystyle \|\nabla \mathbf {u} \|\ll 1} , it is possible to perform a geometric linearization of any one of the finite strain tensors used in finite strain theory, e.g. the Lagrangian finite strain tensor E {\displaystyle \mathbf {E} } , and the Eulerian finite strain tensor e {\displaystyle \mathbf {e} } . In such a linearization, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have
E = 1 2 ( ∇ X u + ( ∇ X u ) T + ( ∇ X u ) T ∇ X u ) ≈ 1 2 ( ∇ X u + ( ∇ X u ) T ) {\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)}
or
E K L = 1 2 ( ∂ U K ∂ X L + ∂ U L ∂ X K + ∂ U M ∂ X K ∂ U M ∂ X L ) ≈ 1 2 ( ∂ U K ∂ X L + ∂ U L ∂ X K ) {\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)}
and
e = 1 2 ( ∇ x u + ( ∇ x u ) T − ∇ x u ( ∇ x u ) T ) ≈ 1 2 ( ∇ x u + ( ∇ x u ) T ) {\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)}
or
e r s = 1 2 ( ∂ u r ∂ x s + ∂ u s ∂ x r − ∂ u k ∂ x r ∂ u k ∂ x s ) ≈ 1 2 ( ∂ u r ∂ x s + ∂ u s ∂ x r ) {\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)}
This linearization implies that the Lagrangian description and the Eulerian description are approximately the same as there is little difference in the material and spatial coordinates of a given material point in the continuum. Therefore, the material displacement gradient tensor components and the spatial displacement gradient tensor components are approximately equal. Thus we have
E ≈ e ≈ ε = 1 2 ( ( ∇ u ) T + ∇ u ) {\displaystyle \mathbf {E} \approx \mathbf {e} \approx {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left((\nabla \mathbf {u} )^{T}+\nabla \mathbf {u} \right)}
or
E K L ≈ e r s ≈ ε i j = 1 2 ( u i , j + u j , i ) {\displaystyle E_{KL}\approx e_{rs}\approx \varepsilon _{ij}={\frac {1}{2}}\left(u_{i,j}+u_{j,i}\right)}
where ε i j {\displaystyle \varepsilon _{ij}} are the components of the infinitesimal strain tensor ε {\displaystyle {\boldsymbol {\varepsilon }}} , also called Cauchy's strain tensor, linear strain tensor, or small strain tensor.
ε i j = 1 2 ( u i , j + u j , i ) = [ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ] = [ ∂ u 1 ∂ x 1 1 2 ( ∂ u 1 ∂ x 2 + ∂ u 2 ∂ x 1 ) 1 2 ( ∂ u 1 ∂ x 3 + ∂ u 3 ∂ x 1 ) 1 2 ( ∂ u 2 ∂ x 1 + ∂ u 1 ∂ x 2 ) ∂ u 2 ∂ x 2 1 2 ( ∂ u 2 ∂ x 3 + ∂ u 3 ∂ x 2 ) 1 2 ( ∂ u 3 ∂ x 1 + ∂ u 1 ∂ x 3 ) 1 2 ( ∂ u 3 ∂ x 2 + ∂ u 2 ∂ x 3 ) ∂ u 3 ∂ x 3 ] {\displaystyle {\begin{aligned}\varepsilon _{ij}&={\frac {1}{2}}\left(u_{i,j}+u_{j,i}\right)\\&={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\\\end{bmatrix}}\\&={\begin{bmatrix}{\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{bmatrix}}\end{aligned}}}
or using different notation:
[ ε x x ε x y ε x z ε y x ε y y ε y z ε z x ε z y ε z z ] = [ ∂ u x ∂ x 1 2 ( ∂ u x ∂ y + ∂ u y ∂ x ) 1 2 ( ∂ u x ∂ z + ∂ u z ∂ x ) 1 2 ( ∂ u y