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Green & Almansi Strains

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Introduction

The previous page on small strains demonstrated that their actual limitation is not small strains at all, but rather small rotations. It was demonstrated that as the amount of rotation grows, so does the inaccuracies in the small strain tensor.

It was also demonstrated that the stretch tensor, specifically \({\bf U} - {\bf I}\), fulfills all the desired properties of a strain tensor and is not limited to small rotations. However, \({\bf U}\) is very difficult to compute (see this page on polar decompositions), so an alternative strain definition is needed that is easy to calculate and is not corrupted by rigid body rotations. The answer to this dilemma is the Green strain tensor.

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Green Strain Definition

The Green strain tensor, \({\bf E}\), is based on the deformation gradient as follows.

\[ {\bf E} = {1 \over 2} \left( {\bf F}^T \cdot {\bf F} - {\bf I} \right) \]
Recall that \({\bf F}^T \cdot {\bf F}\) completely eliminates the rigid body rotation, \({\bf R}\), from the problem because

\[ {\bf F}^T \! \cdot {\bf F} \quad = \quad \left( {\bf R} \cdot {\bf U} \right)^T \cdot \left( {\bf R} \cdot {\bf U} \right) \quad = \quad {\bf U}^T \! \cdot {\bf R}^T \cdot {\bf R} \cdot {\bf U} \quad = \quad {\bf U}^T \! \cdot {\bf U} \]
And this leaves only the stretch tensor, \({\bf U}\), in the calculation, confirming that the result is indeed independent of any rotations (you will get the same computed strain regardless of the amount of rotation).

The Green strain tensor, in tensor notation, is

\[ E_{ij} = {1 \over 2} \left( F_{ki} F_{kj} - \delta_{ij} \right) \]
and since \(F_{ij} = \delta_{ij} + u_{i,j}\), then

\[ \begin{eqnarray} E_{ij} & = & {1 \over 2} \left[ (\delta_{ki} + u_{k,i}) (\delta_{kj} + u_{k,j}) - \delta_{ij} \right] \\ \\ \\ & = & {1 \over 2} \left[ \delta_{ki} \delta_{kj} + \delta_{ki} u_{k,j} + \delta_{kj} u_{k,i} + u_{k,i} u_{k,j} - \delta_{ij} \right] \\ \\ \\ & = & {1 \over 2} \left[ \delta_{ij} + u_{i,j} + u_{j,i} + u_{k,i} u_{k,j} - \delta_{ij} \right] \\ \\ \\ & = & {1 \over 2} \left[ u_{i,j} + u_{j,i} + u_{k,i} u_{k,j} \right] \end{eqnarray} \]
This could be written more explicitly as

\[ E_{ij} = {1 \over 2} \left( {\partial \, u_i \over \partial X_j} + {\partial \, u_j \over \partial X_i} + {\partial \, u_k \over \partial X_i} {\partial \, u_k \over \partial X_j} \right) \]
The strain tensor, which is symmetric, is

\[ {\bf E} = \left[ \matrix{ E_{xx} & E_{xy} & E_{xz} \\ \\ E_{xy} & E_{yy} & E_{yz} \\ \\ E_{xz} & E_{yz} & E_{zz} } \right] \]
The specific expressions for each component of the Green strain tensor are listed here. Keep in mind that \({\bf u} = (u,v,w)\).

\[ \begin{eqnarray} E_{xx} & = & {\partial \, u \over \partial X} + {1 \over 2} \left[ \left( {\partial \, u \over \partial X} \right)^2 + \left( {\partial \, v \over \partial X} \right)^2 + \left( {\partial \, w \over \partial X} \right)^2 \right] \\ \\ \\ E_{yy} & = & {\partial \, v \over \partial Y} + {1 \over 2} \left[ \left( {\partial \, u \over \partial Y} \right)^2 + \left( {\partial \, v \over \partial Y} \right)^2 + \left( {\partial \, w \over \partial Y} \right)^2 \right] \\ \\ \\ E_{zz} & = & {\partial \, w \over \partial Z} + {1 \over 2} \left[ \left( {\partial \, u \over \partial Z} \right)^2 + \left( {\partial \, v \over \partial Z} \right)^2 + \left( {\partial \, w \over \partial Z} \right)^2 \right] \\ \\ \\ E_{xy} & = & {1 \over 2} \left( {\partial \, u \over \partial Y} + {\partial \, v \over \partial X} \right) + {1 \over 2} \left[ {\partial \, u \over \partial X} {\partial \, u \over \partial Y} + {\partial \, v \over \partial X} {\partial \, v \over \partial Y} + {\partial \, w \over \partial X} {\partial \, w \over \partial Y} \right] \\ \\ \\ E_{xz} & = & {1 \over 2} \left( {\partial \, u \over \partial Z} + {\partial \, w \over \partial X} \right) + {1 \over 2} \left[ {\partial \, u \over \partial X} {\partial \, u \over \partial Z} + {\partial \, v \over \partial X} {\partial \, v \over \partial Z} + {\partial \, w \over \partial X} {\partial \, w \over \partial Z} \right] \\ \\ \\ E_{yz} & = & {1 \over 2} \left( {\partial \, v \over \partial Z} + {\partial \, w \over \partial Y} \right) + {1 \over 2} \left[ {\partial \, u \over \partial Y} {\partial \, u \over \partial Z} + {\partial \, v \over \partial Y} {\partial \, v \over \partial Z} + {\partial \, w \over \partial Y} {\partial \, w \over \partial Z} \right] \\ \end{eqnarray} \]
The equations are certainly too complex to provide much intuitive insight into their properties, other than the fact that it is known that they are independent of rotations. Perhaps the best insight is that they can be grouped into...

\[ \text{Green Strain = Small Strain Terms + Quadratic Terms} \]
The small strain terms are the same, possessing all the desirable properties of engineering strain behavior. The quadratic terms are what gives the Green strain tensor its rotation independence. But this does come at a price, the \(\epsilon = \Delta L / L_o\) and \(\gamma = D / T\) behaviors are affected by the quadratic terms when the strains are large. (Not just rotations this time, but actual strains.) This is discussed in more detail shortly.
The examples above make it clear that the Green strain tensor has several interesting properties that are usually considered desirable: independence from rotations, normal stretching due to shear, etc. But the compromise is that simple uniaxial tension is not exactly equal to \(\Delta L / L_o\) at large strains.

An Alternative Derivation

The Green strain is often presented in textbooks in a way that does not highlight its rotational independence, but instead in a way that I feel is more coincidental than physical.

Since \({\bf F}\) is defined as

\[ {\bf F} = {\partial \, {\bf x} \over \partial {\bf X}} \]
This means that it can be used to relate an initial undeformed differential length, \({\bf dX}\), to its deformed result, \({\bf dx}\), as follows.

\[ {\bf dx} = {\bf F} \cdot {\bf dX} \qquad \text{because} \qquad {\bf dx} = {\partial \, {\bf x} \over \partial {\bf X}} \cdot {\bf dX} \]
The above relationship is exploited in the following story leading to a derivation of the Green strain tensor. It goes like this...

Since \({\bf dx} = {\bf F} \cdot {\bf dX}\), then

\[ {\bf dx} \cdot {\bf dx} = {\bf F} \cdot {\bf dX} \cdot {\bf F} \cdot {\bf dX} \]
Tensor notation is needed here to figure out how to manipulate this result. So in tensor notation, it is

\[ \begin{eqnarray} dx_i dx_i & = & F_{ij} dX_j F_{ik} dX_k \\ \\ & = & dX_j F_{ij} F_{ik} dX_k \\ \end{eqnarray} \]
Which can be recognized as containing a \({\bf F}^T \! \cdot {\bf F}\) term. So this means

\[ \begin{eqnarray} {\bf dx} \cdot {\bf dx} & = & {\bf F} \cdot {\bf dX} \cdot {\bf F} \cdot {\bf dX} \\ \\ & = & {\bf dX} \cdot ( {\bf F}^T \! \cdot {\bf F} ) \cdot {\bf dX} \end{eqnarray} \]
Now subtract \({\bf dX} \cdot {\bf dX}\) from both sides

\[ \begin{eqnarray} {\bf dx} \cdot {\bf dx} - {\bf dX} \cdot {\bf dX} & = & {\bf dX} \cdot ( {\bf F}^T \! \cdot {\bf F} ) \cdot {\bf dX} - {\bf dX} \cdot {\bf dX} \\ \\ & = & {\bf dX} \cdot ( {\bf F}^T \! \cdot {\bf F} ) \cdot {\bf dX} - {\bf dX} \cdot {\bf I} \cdot {\bf dX} \\ \\ & = & {\bf dX} \cdot ( {\bf F}^T \! \cdot {\bf F} - {\bf I}) \cdot {\bf dX} \end{eqnarray} \]
But \({\bf F}^T \! \cdot {\bf F} - {\bf I}\) equals \(2 {\bf E}\). So

\[ {\bf dx} \cdot {\bf dx} - {\bf dX} \cdot {\bf dX} = {\bf dX} \cdot ( 2 {\bf E} ) \cdot {\bf dX} \]
This is used to justify that

\[ E = {ds^2 - dS^2 \over 2 \, dS^2 } \]
where \(ds^2\) is the deformed length squared, and \(dS^2\) is the undeformed length squared.

This is often presented as if it is somehow a better definition of strain than the "boring old" \(\Delta L / L_o\) relationship. But I consider this to only be a coincidence. It is not better at all.

It can be checked by substituting \(L_o\) for \(dS\), and \(L_o + \Delta L\) for \(ds\). It will lead to the now familiar relationship

\[ E \;\; = \;\; {ds^2 - dS^2 \over 2 \, dS^2} \;\; = \;\; {(L_o + \Delta L)^2 - L_o^2 \over 2 \, L_o^2} \;\; = \;\; {\Delta L \over L_o} + {1 \over 2} \left( {\Delta L \over L_o} \right)^2 \]

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