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Displacement and Strain: The Displacement Gradient Tensor

Another three dimensional measure of deformation is the displacement gradient tensor. The displacement gradient tensor appears naturally when we attempt to write the relationship between a tangent vector dX in the reference configuration deformation and its image under deformation dx such that:

    \[ dx=dX + du \]

where du is the “displacement” vector that describes the change in tangent vectors.

As discussed in the deformation gradient section, dx and dX are related as follows:

    \[ dx=FdX \]

Therefore, the “displacement” vector du can be written as:

    \[ du=dx - dX = (F-I)dX \]

The tensor \nabla u=F-I is denoted the displacement gradient tensor and can be written in component form as follows:

    \[ \nabla u= \left(\begin{array}{ccc} \frac{\partial u_1}{\partial X_1} & \frac{\partial u_1}{\partial X_2} & \frac{\partial u_1}{\partial X_3}\\ \frac{\partial u_2}{\partial X_1} & \frac{\partial u_2}{\partial X_2} & \frac{\partial u_2}{\partial X_3}\\ \frac{\partial u_3}{\partial X_1} & \frac{\partial u_3}{\partial X_2} & \frac{\partial u_3}{\partial X_3} \end{array}\right) = \left(\begin{array}{ccc} \frac{\partial x_1}{\partial X_1} & \frac{\partial x_1}{\partial X_2} & \frac{\partial x_1}{\partial X_3}\\ \frac{\partial x_2}{\partial X_1} & \frac{\partial x_2}{\partial X_2} & \frac{\partial x_2}{\partial X_3}\\ \frac{\partial x_3}{\partial X_1} & \frac{\partial x_3}{\partial X_2} & \frac{\partial x_3}{\partial X_3} \end{array}\right) - \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right) \]

As described in the skewsymmetric tensors section, every tensor can be uniquely decomposed into two additive components, a symmetric tensor and a skewsymmetric tensor. By denoting the symmetric part as \varepsilon or the “infinitesimal strain tensor” and the skewsymmetric part as W_{inf} or the “infintesimal rotation tensor” we can write the relationship between the vectors in the reference and deformed configuration as follows:

    \[ dx=dX + \nabla u dX = dX + \varepsilon dX + W_{inf} dX  \]

In other words, the additive decomposition of the displacement gradient tensor allows to write the deformed vector dx as the additive combination of three vectors: the original vector dX, plus a “strain” or “stretch” component \varepsilon dX, plus a “rotation” component W_{inf} dX. The stretch component can be calculated using the symmetric tensor \varepsilon while the rotation component can be calculated using the skewsymmetric tensor W_{inf}. Both tensors are physically meaningful when \nabla u has very small components \frac{\partial u_i}{\partial X_j}<<1 (small displacements).

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