(Solved): Consider the following strain-energy function per unit volume, which represents a transversely isot ...

Consider the following strain-energy function per unit volume, which represents a transversely isotropic incompressible material: $?\left({\mathrm{I}}_1,{\mathrm{I}}_4\right)={?}_1\left({\mathrm{I}}_1?3\right)+\frac{{?}_2}{2}\left(e^{{?}_3{\left({\mathrm{I}}_4?1\right)}^2}?1\right)$ where ${\mathrm{I}}_1$ is the first invariant of $\mathbf{C},{\mathrm{I}}_4$ is the pseudo-invariant associated with $\mathbf{C}$ and the fiber direction ${\mathbf{a}}_0$, and ${?}_1,{?}_2,{?}_3$ are material constants. a) Using a fiber direction of ${\mathbf{a}}_0={\mathbf{E}}_1$, define the $2^{\text{nd }}$ Piola-Kirchhoff stress tensor for this material. (don't forget to enforce incompressibility) To continue we wish to analyze a thin sheet of this material (see the figure below). We will assume that the sheet of material is thin enough to assume a plane stress condition. Note that plane stress implies: - the stress components ${\mathrm{S}}_{13}={\mathrm{S}}_{31}={\mathrm{S}}_{23}={\mathrm{S}}_{32}={\mathrm{S}}_{33}=0$ - the components of the right Cauchy-Green deformation tensor ${\mathrm{C}}_{13}={\mathrm{C}}_{31}={\mathrm{C}}_{23}={\mathrm{C}}_{32}=0$ Using the information above please solve the following: b) Show that for the case of plane stress and incompressibility ${\mathrm{C}}_{33}$ can be expressed in terms of the remaining components of $\mathbf{C}$. c) Additionally, show that for the case of plane stress and incompressibility the Lagrange multiplier p can be defined explicitly in terms of material constants and the components of C. d) Say that the sheet of material undergoes a planar equi-biaxial extension test, which is represented by the motion $x_1=?{\mathrm{X}}_1,x_2=?{\mathrm{X}}_2,x_3={?}_3{\mathrm{X}}_3$ where $?$ is prescribed and ${?}_3$ is determined from incompressibility. Develop expressions for the non-zero components of $\mathbf{S}$ only in terms of $?$ and the material constants.