# What is the gradient of deformation gradient $F$?

Deformation gradient is defined as $$F_{iJ}=\frac{\partial x_i}{\partial X_J},\;\mathbf{F}=\frac{\partial\mathbf{x}}{\partial\mathbf{X}},$$ where $$\mathbf{x}$$ is spatial coordinates; $$\mathbf{X}$$ is material coordinates. I wonder whether $$\nabla\mathbf{F}$$ has physical meaning. Here is my derivation: $$\nabla\mathbf{F}=\frac{\partial\mathbf{F}}{\partial\mathbf{x}}=\frac{\partial}{\partial\mathbf{x}}\frac{\partial\mathbf{x}}{\partial\mathbf{X}},$$ where $$\nabla$$ is the del operator with respect to spatial coordinates. Then we interchange the differential order on the right-hand side, $$\nabla\mathbf{F}=\frac{\partial}{\partial\mathbf{X}}\frac{\partial\mathbf{x}}{\partial\mathbf{x}}=\frac{\partial\mathbf{I}}{\partial\mathbf{X}},$$ where $$\mathbf{I}$$ is a second-order unit tensor. Now we have $$\nabla\mathbf{F}=\nabla_0\mathbf{I},$$ where $$\nabla_0$$ is the del operator with respect to material coordinates. The answer seems to be zero if there is no mistake in my derivation, but what is its physical meaning?

The deformation is the application $$\begin{array}{rl} \boldsymbol y:\Omega_0 &\longmapsto\Omega\\ \boldsymbol X &\longmapsto \boldsymbol y(\boldsymbol X) \end{array}$$ where $$\Omega_0$$ and $$\Omega$$ designate the initial (material) and current (spatial) configurations, respectively.
The deformation gradient has components $$F_{i\alpha}=(\mathrm{grad}\,\boldsymbol y)_{i\alpha}=\frac{\partial y_i}{\partial X_\alpha}$$ and is also a function of the material coordinates $$\boldsymbol F(\boldsymbol X)$$.
The gradient of the deformation gradient naturally reads $$(\mathrm{grad}\,\boldsymbol F)_{i\alpha\beta}=\frac{\partial^2 y_i}{\partial X_\alpha\partial X_\beta}.$$ The deformation mapping is not a function of the spatial coordinates, hence they have no reason to appear at any step.
• It is preferable not to mix up the spacial variable $$\boldsymbol x$$ (an independent variable in $$\Omega$$) with the deformation $$\boldsymbol y$$ (a mapping from $$\Omega_0$$ to $$\Omega$$). In OP's suggestion, the identity tensor emerges precisely because of this confusion.