Displacement gradient ($H=\nabla \mathbf{u}$)

For displacement $u(X)$ in reference coordinates, the displacement gradient is $H_{ij}=\partial u_i/\partial X_j$. It records how nearby points move differently. The deformation gradient is $F=I+H$.

In small-strain kinematics, $ϵ=(H+H^T)/2$ describes strain, while the antisymmetric part gives infinitesimal rotation. For a rigid translation, $H=0$. For $u=(\alpha X_1,0,0)$, $H_{11}=\alpha$ describes uniform axial extension. For finite rotations, the linearized strain is not an exact rotation-invariant measure, so a finite-deformation strain tensor is needed.