## 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 in the reference configuration deformation and its image under deformation such that:

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

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

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

The tensor is denoted the displacement gradient tensor and can be written in component form as follows:

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 or the “infinitesimal strain tensor” and the skewsymmetric part as or the “infintesimal rotation tensor” we can write the relationship between the vectors in the reference and deformed configuration as follows:

In other words, the additive decomposition of the displacement gradient tensor allows to write the deformed vector as the additive combination of three vectors: the original vector , plus a “strain” or “stretch” component , plus a “rotation” component . The stretch component can be calculated using the symmetric tensor while the rotation component can be calculated using the skewsymmetric tensor . Both tensors are physically meaningful when has very small components (small displacements).