Finite Strain Theory - Finite Strain Tensors

Finite Strain Tensors

The concept of strain is used to evaluate how much a given displacement differs locally from a rigid body displacement (Ref. Lubliner). One of such strains for large deformations is the Lagrangian finite strain tensor, also called the Green-Lagrangian strain tensor or Green – St-Venant strain tensor, defined as

or as a function of the displacement gradient tensor

The Green-Lagrangian strain tensor is a measure of how much differs from . It can be shown that this tensor is a special case of a general formula for Lagrangian strain tensors (Hill 1968):

For different values of we have:

$\begin{align} \mathbf E_{(1)}&=\frac{1}{2}(\mathbf U^{2}- \mathbf I) \qquad \text{Green-Lagrangian strain tensor}\\ \mathbf E_{(1/2)}&=(\mathbf U- \mathbf I) \qquad \text{Biot strain tensor}\\ \mathbf E_{(0)}&=\ln \mathbf U \qquad \text{Logarithmic strain, Natural strain, True strain, or Hencky strain} \end{align}\,\!$

The Eulerian-Almansi finite strain tensor, referenced to the deformed configuration, i.e. Eulerian description, is defined as

$\mathbf e=\frac{1}{2}(\mathbf I - \mathbf c) \qquad \text{or} \qquad e_{rs}=\frac{1}{2}\left(\delta_{rs} - \frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s} \right)\,\!$

or as a function of the displacement gradients we have

Derivation of the Lagrangian and Eulerain finite strain tensors
A measure of deformation is the difference between the squares of the differential line element, in the undeformed configuration, and, in the deformed configuration (Figure 2). Deformation has occurred if the difference is non zero, otherwise a rigid-body displacement has occurred. Thus we have, $d\mathbf{x}^2 - d\mathbf{X}^2=d\mathbf x\cdot d\mathbf x-d\mathbf X\cdot d\mathbf X \qquad \text{or} \qquad (dx)^2 - (dX)^2=dx_jdx_j-dX_M\,dX_M\,\!$ In the Lagrangian description, using the material coordinates as the frame of reference, the linear transformation between the differential lines is Then we have, $\begin{align} d\mathbf{x}^2&=d\mathbf x \cdot d\mathbf x \\ &= \mathbf F \cdot d\mathbf X \cdot \mathbf F \cdot d\mathbf X \\ &= d\mathbf X \cdot \mathbf F^T\mathbf F \cdot d\mathbf X \\ &= d\mathbf X\cdot\mathbf C\cdot d\mathbf X \end{align} \qquad \text{or} \qquad \begin{align} (dx)^2&=dx_j\,dx_j \\ &= \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}\,dX_K\,dX_L \\ &= C_{KL}\,dX_K\,dX_L \\ \end{align}\,\!$ where are the components of the right Cauchy-Green deformation tensor, . Then, replacing this equation into the first equation we have, $\begin{align} d\mathbf{x}^2 - d\mathbf{X}^2 &= d\mathbf X\cdot\mathbf C\cdot d\mathbf X-d\mathbf X\cdot d\mathbf X \\ &=d\mathbf X\cdot (\mathbf C - \mathbf I)\cdot d\mathbf X \\ &= d\mathbf X \cdot 2\mathbf E \cdot d\mathbf X \\ \end{align}\,\!$ or $\begin{align} (dx)^2 - (dX)^2 &= \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}\,dX_K\,dX_L-dX_M\,dX_M \\ &= \left( \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}-\delta_{KL}\right)\,dX_K\,dX_L \\ &=2E_{KL}\,dX_K\,dX_L \end{align}\,\!$ where, are the components of a second-order tensor called the Green – St-Venant strain tensor or the Lagrangian finite strain tensor, In the Eulerian description, using the spatial coordinates as the frame of reference, the linear transformation between the differential lines is where are the components of the spatial deformation gradient tensor, . Thus we have $\begin{align} d\mathbf{X}^2&=d\mathbf X \cdot d\mathbf X \\ &= \mathbf F^{-1} \cdot d\mathbf x \cdot \mathbf F^{-1} \cdot d\mathbf x \\ &= d\mathbf x \cdot \mathbf F^{-T}\mathbf F^{-1} \cdot d\mathbf x \\ &= d\mathbf x\cdot\mathbf c\cdot d\mathbf x \end{align} \qquad \text{or} \qquad \begin{align} (dX)^2&=dX_M\,dX_M \\ &= \frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s}\,dx_r\,dx_s \\ &= c_{rs}\,dx_r\,dx_s \\ \end{align}\,\!$ where the second order tensor is called Cauchy's deformation tensor, . Then we have, $\begin{align} d\mathbf{x}^2 - d\mathbf{X}^2 &= d\mathbf x\cdot d\mathbf x-d\mathbf x\cdot\mathbf c\cdot d\mathbf x \\ &=d\mathbf x\cdot (\mathbf I - \mathbf c)\cdot d\mathbf x \\ &= d\mathbf x \cdot 2\mathbf e \cdot d\mathbf x \\ \end{align}\,\!$ or $\begin{align} (dx)^2 - (dX)^2 &= dx_j\,dx_j-\frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s}\,dx_r\,dx_s \\ &= \left(\delta_{rs} - \frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s} \right)\,dx_r\,dx_s \\ &=2e_{rs}\,dx_r\,dx_s \end{align}\,\!$ where, are the components of a second-order tensor called the Eulerian-Almansi finite strain tensor, $\mathbf e=\frac{1}{2}(\mathbf I - \mathbf c) \qquad \text{or} \qquad e_{rs}=\frac{1}{2}\left(\delta_{rs} - \frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s} \right)\,\!$ Both the Lagrangian and Eulerian finite strain tensors can be conveniently expressed in terms of the displacement gradient tensor. For the Lagrangian strain tensor, first we differentiate the displacement vector with respect to the material coordinates to obtain the material displacement gradient tensor, $\begin{align} \mathbf u(\mathbf X,t) &= \mathbf x(\mathbf X,t) - \mathbf X \\ \nabla_{\mathbf X}\mathbf u &= \mathbf F - \mathbf I \\ \mathbf F &= \nabla_{\mathbf X}\mathbf u + \mathbf I \\ \end{align} \qquad \text{or} \qquad \begin{align} u_i& = x_i-\delta_{iJ}X_J \\ \delta_{iJ}U_J &= x_i-\delta_{iJ}X_J \\ x_i&=\delta_{iJ}\left(U_J+X_J\right) \\ \frac{\partial x_i}{\partial X_K}&=\delta_{iJ}\left(\frac{\partial U_J}{\partial X_K}+\delta_{JK}\right) \\ \end{align} \,\!$ Replacing this equation into the expression for the Lagrangian finite strain tensor we have $\begin{align} \mathbf E &= \frac{1}{2}\left(\mathbf F^T\mathbf F-\mathbf I\right) \\ &=\frac{1}{2}\left \\ &=\frac{1}{2}\left \\ \end{align}\,\!$ or $\begin{align} E_{KL}&=\frac{1}{2}\left( \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}-\delta_{KL}\right) \\ &=\frac{1}{2}\left \\ &=\frac{1}{2}\left \\ &=\frac{1}{2}\left \\ &=\frac{1}{2}\left(\frac{\partial U_K}{\partial X_L}+\frac{\partial U_L}{\partial X_K}+\frac{\partial U_M}{\partial X_K}\frac{\partial U_M}{\partial X_L}\right) \end{align}\,\!$ Similarly, the Eulerian-Almansi finite strain tensor can be expressed as

Derivation of the Lagrangian and Eulerain finite strain tensors

A measure of deformation is the difference between the squares of the differential line element, in the undeformed configuration, and, in the deformed configuration (Figure 2). Deformation has occurred if the difference is non zero, otherwise a rigid-body displacement has occurred. Thus we have,

$d\mathbf{x}^2 - d\mathbf{X}^2=d\mathbf x\cdot d\mathbf x-d\mathbf X\cdot d\mathbf X \qquad \text{or} \qquad (dx)^2 - (dX)^2=dx_jdx_j-dX_M\,dX_M\,\!$

In the Lagrangian description, using the material coordinates as the frame of reference, the linear transformation between the differential lines is

Then we have,

$\begin{align} d\mathbf{x}^2&=d\mathbf x \cdot d\mathbf x \\ &= \mathbf F \cdot d\mathbf X \cdot \mathbf F \cdot d\mathbf X \\ &= d\mathbf X \cdot \mathbf F^T\mathbf F \cdot d\mathbf X \\ &= d\mathbf X\cdot\mathbf C\cdot d\mathbf X \end{align} \qquad \text{or} \qquad \begin{align} (dx)^2&=dx_j\,dx_j \\ &= \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}\,dX_K\,dX_L \\ &= C_{KL}\,dX_K\,dX_L \\ \end{align}\,\!$

where are the components of the right Cauchy-Green deformation tensor, . Then, replacing this equation into the first equation we have,

$\begin{align} d\mathbf{x}^2 - d\mathbf{X}^2 &= d\mathbf X\cdot\mathbf C\cdot d\mathbf X-d\mathbf X\cdot d\mathbf X \\ &=d\mathbf X\cdot (\mathbf C - \mathbf I)\cdot d\mathbf X \\ &= d\mathbf X \cdot 2\mathbf E \cdot d\mathbf X \\ \end{align}\,\!$

$\begin{align} (dx)^2 - (dX)^2 &= \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}\,dX_K\,dX_L-dX_M\,dX_M \\ &= \left( \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}-\delta_{KL}\right)\,dX_K\,dX_L \\ &=2E_{KL}\,dX_K\,dX_L \end{align}\,\!$

where, are the components of a second-order tensor called the Green – St-Venant strain tensor or the Lagrangian finite strain tensor,

In the Eulerian description, using the spatial coordinates as the frame of reference, the linear transformation between the differential lines is

where are the components of the spatial deformation gradient tensor, . Thus we have

$\begin{align} d\mathbf{X}^2&=d\mathbf X \cdot d\mathbf X \\ &= \mathbf F^{-1} \cdot d\mathbf x \cdot \mathbf F^{-1} \cdot d\mathbf x \\ &= d\mathbf x \cdot \mathbf F^{-T}\mathbf F^{-1} \cdot d\mathbf x \\ &= d\mathbf x\cdot\mathbf c\cdot d\mathbf x \end{align} \qquad \text{or} \qquad \begin{align} (dX)^2&=dX_M\,dX_M \\ &= \frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s}\,dx_r\,dx_s \\ &= c_{rs}\,dx_r\,dx_s \\ \end{align}\,\!$

where the second order tensor is called Cauchy's deformation tensor, . Then we have,

$\begin{align} d\mathbf{x}^2 - d\mathbf{X}^2 &= d\mathbf x\cdot d\mathbf x-d\mathbf x\cdot\mathbf c\cdot d\mathbf x \\ &=d\mathbf x\cdot (\mathbf I - \mathbf c)\cdot d\mathbf x \\ &= d\mathbf x \cdot 2\mathbf e \cdot d\mathbf x \\ \end{align}\,\!$

$\begin{align} (dx)^2 - (dX)^2 &= dx_j\,dx_j-\frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s}\,dx_r\,dx_s \\ &= \left(\delta_{rs} - \frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s} \right)\,dx_r\,dx_s \\ &=2e_{rs}\,dx_r\,dx_s \end{align}\,\!$

where, are the components of a second-order tensor called the Eulerian-Almansi finite strain tensor,

$\mathbf e=\frac{1}{2}(\mathbf I - \mathbf c) \qquad \text{or} \qquad e_{rs}=\frac{1}{2}\left(\delta_{rs} - \frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s} \right)\,\!$

Both the Lagrangian and Eulerian finite strain tensors can be conveniently expressed in terms of the displacement gradient tensor. For the Lagrangian strain tensor, first we differentiate the displacement vector with respect to the material coordinates to obtain the material displacement gradient tensor,

$\begin{align} \mathbf u(\mathbf X,t) &= \mathbf x(\mathbf X,t) - \mathbf X \\ \nabla_{\mathbf X}\mathbf u &= \mathbf F - \mathbf I \\ \mathbf F &= \nabla_{\mathbf X}\mathbf u + \mathbf I \\ \end{align} \qquad \text{or} \qquad \begin{align} u_i& = x_i-\delta_{iJ}X_J \\ \delta_{iJ}U_J &= x_i-\delta_{iJ}X_J \\ x_i&=\delta_{iJ}\left(U_J+X_J\right) \\ \frac{\partial x_i}{\partial X_K}&=\delta_{iJ}\left(\frac{\partial U_J}{\partial X_K}+\delta_{JK}\right) \\ \end{align} \,\!$

Replacing this equation into the expression for the Lagrangian finite strain tensor we have

$\begin{align} \mathbf E &= \frac{1}{2}\left(\mathbf F^T\mathbf F-\mathbf I\right) \\ &=\frac{1}{2}\left \\ &=\frac{1}{2}\left \\ \end{align}\,\!$

$\begin{align} E_{KL}&=\frac{1}{2}\left( \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}-\delta_{KL}\right) \\ &=\frac{1}{2}\left \\ &=\frac{1}{2}\left \\ &=\frac{1}{2}\left \\ &=\frac{1}{2}\left(\frac{\partial U_K}{\partial X_L}+\frac{\partial U_L}{\partial X_K}+\frac{\partial U_M}{\partial X_K}\frac{\partial U_M}{\partial X_L}\right) \end{align}\,\!$

Similarly, the Eulerian-Almansi finite strain tensor can be expressed as

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