Infinitesimal Strain Theory - Infinitesimal Strain Tensor

Infinitesimal Strain Tensor

For infinitesimal deformations of a continuum body, in which the displacements and the displacement gradients are small compared to unity, i.e., and, it is possible to perform a geometric linearisation of the Lagrangian finite strain tensor, and the Eulerian finite strain tensor . In such a linearisation, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have

or

and

or

This linearisation 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 components and the spatial displacement gradient components are approximately equal. Thus we have

or

where are the components of the infinitesimal strain tensor, also called Cauchy's strain tensor, linear strain tensor, or small strain tensor.

\begin{align} \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{align}

or using different notation:

\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] \,\!

Furthermore, since the deformation gradient can be expressed as where is the second-order identity tensor, we have

Also, from the general expression for the Lagrangian and Eulerian finite strain tensors we have

\begin{align} \mathbf E_{(m)}& =\frac{1}{2m}(\mathbf U^{2m}-\boldsymbol{I}) = \frac{1}{2m} \approx \frac{1}{2m}\approx \boldsymbol{\varepsilon}\\ \mathbf e_{(m)}& =\frac{1}{2m}(\mathbf V^{2m}-\boldsymbol{I})= \frac{1}{2m}\approx \boldsymbol{\varepsilon} \end{align}

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