(Solved): The Rivlin-Ericksen tensors A1,A2, are defined by Ai=FTC(i)F1,i=1,2, where C(i ...

The Rivlin-Ericksen tensors ${\mathbf{A}}_1,{\mathbf{A}}_2,\dots$ are defined by ${\mathbf{A}}_i={\mathbf{F}}^{?\mathrm{T}}{\mathbf{C}}^{(i)}{\mathbf{F}}^{?1},i=1,2,\dots$ where ${\mathbf{C}}^{(i)}$ represents the ${\mathrm{i}}^{\text{th }}$ material time derivative of $\mathbf{C}$, i.e. ${\mathbf{C}}^{(1)}=\stackrel{?}{\mathbf{C}},{\mathbf{C}}^{(2)}=\ddot{\mathbf{C}}$, etc. Show that: (a) The $1^{\text{st }}$ Rivlin-Eriksen tensor is given by ${\mathbf{A}}_1=2\mathbf{D}$ (b) All Rivlin-Eriksen tensors are obtained recursively from the previous one, i.e. ${\mathbf{A}}_{i+1}={\mathbf{A}}_i\text{ for }i=1,2,\dots$ where this superscript symbol (little ball above) represents the lower convected material time derivative given by $\stackrel{?}{\mathbf{T}}=\stackrel{?}{\mathbf{T}}+{\mathbf{L}}^{?}\mathbf{T}+\mathbf{TL}$ (this is an time derivative that is objective) (c) All Rivlin-Ericksen tensors are symmetric (great stuff, guarantees real eigenvalues) (d) All Rivlin-Ericksen tensors are objective (great stuff to define constitutive equations)