Deformation map ($\mathbf{x}=\mathbf{\chi }(\mathbf{X},t)$)

A deformation map sends each particle's reference configuration position to its current configuration position. Its spatial derivative with respect to the Lagrangian coordinates is the deformation gradient.

A radial deformation map preserving each particle's direction from a common centre. In an orthonormal basis of spherical coordinates, its deformation gradient has principal stretches $f^{\prime }(R),f(R)/R,f(R)/R$. Thus incompressibility gives $f^{\prime }(R)f(R{)}^2=R^2$, and integration gives $f(R{)}^3=R^3+C$.