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The upper-convected derivative of a contravariant tensor is $\stackrel{▽}{\mathbf{C}}=D\mathbf{C}/Dt-(\nabla \mathbf{u})\mathbf{C}-\mathbf{C}(\nabla \mathbf{u}{)}^T$.

Differentiate the Kirchhoff stress tensor relation $\mathbf{\tau }=FS{F}^T$, using $\dot{F}=LF$ for the velocity gradient. Subtracting the two deformation terms gives the displayed upper-convected derivative. This removes the transport generated solely by deformation of a fixed material stress tensor.