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3. The motion of a continuum body at a given time is defined as $x_1 = X_1 - AX_3$ $x_2 = X_2 - AX_3$ $x_3 = -AX_1 + AX_2 + X_3$ calculate material deformation gradient tensor $\mathbf{E}(\mathbf{X}, t)$ at same time. By using inverse equa- tions of motion, obtain the spatial deformation gradient tensor $\mathbf{F}^{-1}(\mathbf{x})$ and verify $\mathbf{F}\mathbf{F}^{-1} = \mathbf{I}$. 4. For a homogeneous deformation defined by $x_1 = \alpha X_1 + \beta X_2$ $x_2 = -\beta X_1 + \alpha X_2$ $x_3 = \mu X_3$ where $\alpha$, $\beta$ and $\mu$ are constants, calculate the Lagrangian strain tensor $\mathbf{E}$. Show that if $\alpha = \cos \theta$, $\beta = \sin \theta$ and $\mu = 1$ the strain is zero and the mapping corresponds to a rigid body rotation of magnitude $\theta$ about $X_3$ axis. 5. Given the deformation defined by

          3. The motion of a continuum body at a given time is defined as
$x_1 = X_1 - AX_3$
$x_2 = X_2 - AX_3$
$x_3 = -AX_1 + AX_2 + X_3$
calculate material deformation gradient tensor $\mathbf{E}(\mathbf{X}, t)$ at same time. By using inverse equa-
tions of motion, obtain the spatial deformation gradient tensor $\mathbf{F}^{-1}(\mathbf{x})$ and verify $\mathbf{F}\mathbf{F}^{-1} = \mathbf{I}$.
4. For a homogeneous deformation defined by
$x_1 = \alpha X_1 + \beta X_2$
$x_2 = -\beta X_1 + \alpha X_2$
$x_3 = \mu X_3$
where $\alpha$, $\beta$ and $\mu$ are constants, calculate the Lagrangian strain tensor $\mathbf{E}$. Show that if
$\alpha = \cos \theta$, $\beta = \sin \theta$ and $\mu = 1$ the strain is zero and the mapping corresponds to a rigid
body rotation of magnitude $\theta$ about $X_3$ axis.
5. Given the deformation defined by

3. The motion of a continuum body at a given time is defined as
x1 = X1 - AX3
x2 = X2 - AX3
x3 = -AX1 + AX2 + X3
calculate material deformation gradient tensor 𝐄(𝐗, t) at same time. By using inverse equa-
tions of motion, obtain the spatial deformation gradient tensor 𝐅^-1(𝐱) and verify 𝐅𝐅^-1 = 𝐈.
4. For a homogeneous deformation defined by
x1 = α X1 + β X2
x2 = -β X1 + α X2
x3 = μ X3
where α, β and μ are constants, calculate the Lagrangian strain tensor 𝐄. Show that if
α = cosθ, β = sinθ and μ = 1 the strain is zero and the mapping corresponds to a rigid
body rotation of magnitude θ about X3 axis.
5. Given the deformation defined by

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