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bobinthebox

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I'm studying elasticity from classical Gurtin's book, and my professor gave us the following example, during lecture. Unfortunately, this is not present in our references, so I'm posting it here the beginning of the solution, and I will highlight at the end my questions. First I need to state the problem clearly:> Let us consider a body ##B## which has as reference configuration a cube ##[0,b] \times [0,b] \times [0,h]## of incompressible, Neo-Hookean material, without body forces acting on the body.

In the following, ##S## will be the first Piola-Kirchoff stress tensor, and ##f## will indicate the homogeneous deformation.

Boundary conditions on the lateral surfaces:

- ##S n = 0## (i.e. contact forces at the boundary are 0, i.e. I have no traction)

- - - - - - - - - - - -

Boundary conditions on the bottom surface:

- ##u(0)=f(0)-0 = 0## (i.e. origin is fixed during the deformation)

- ##u_3(p)= (f(p)-p)\cdot e_3 = 0## (No displacement along e_3)

- ##S(-e_3) \cdot e_1 = S(-e_3) \cdot e_2 = 0## (Body is constrained to a frictionless plane)

- - - - - - - - - - - - -

Boundary conditions on the top surface:

- ##u_3(p)=\delta## (we allow displacement of an amount $\delta$ on direction $e_3$

- ##S(e_3) \cdot e_1 = S(e_3) \cdot e_2 = 0##- - - - - - - - - - - - -

**Attacking the problem**

To solve it, we make an ansatz, i.e. that the deformation is going to be homogeneous. The field equation is ##\operatorname{div}(S)=0##, since there are no body forces. Also, since the material is Neo-Hookean, we have ##T=\mu B - \pi I##, and hence the first Piola ##S## is:

$$S = \mu F - \pi F^{-T}$$Here's where I start having big troubles:

Considering the polar decomposition of the deformation gradient ##F##, ##F=RU##, we notice that ##R=I## (**

Also, by invariance w.r.t rotation around ##e_3## we have ##\lambda_1 = \lambda_2 = \lambda##.

In the following, ##S## will be the first Piola-Kirchoff stress tensor, and ##f## will indicate the homogeneous deformation.

Boundary conditions on the lateral surfaces:

- ##S n = 0## (i.e. contact forces at the boundary are 0, i.e. I have no traction)

- - - - - - - - - - - -

Boundary conditions on the bottom surface:

- ##u(0)=f(0)-0 = 0## (i.e. origin is fixed during the deformation)

- ##u_3(p)= (f(p)-p)\cdot e_3 = 0## (No displacement along e_3)

- ##S(-e_3) \cdot e_1 = S(-e_3) \cdot e_2 = 0## (Body is constrained to a frictionless plane)

- - - - - - - - - - - - -

Boundary conditions on the top surface:

- ##u_3(p)=\delta## (we allow displacement of an amount $\delta$ on direction $e_3$

- ##S(e_3) \cdot e_1 = S(e_3) \cdot e_2 = 0##- - - - - - - - - - - - -

**Attacking the problem**

To solve it, we make an ansatz, i.e. that the deformation is going to be homogeneous. The field equation is ##\operatorname{div}(S)=0##, since there are no body forces. Also, since the material is Neo-Hookean, we have ##T=\mu B - \pi I##, and hence the first Piola ##S## is:

$$S = \mu F - \pi F^{-T}$$Here's where I start having big troubles:

**I will write in bold where I can't understand**Considering the polar decomposition of the deformation gradient ##F##, ##F=RU##, we notice that ##R=I## (**

**why is that****?) and hence ##F=U## and so $$F=\lambda_1 e_1 \otimes e_1 + \lambda_2 e_2 \otimes e_2 + \lambda_3 e_3 \otimes e_3$$ where ##\lambda_i## are the principal stretches. Therefore, ##\lambda_3 = \frac{h+\delta}{h}## (****why this ratio**?**)Also, by invariance w.r.t rotation around ##e_3## we have ##\lambda_1 = \lambda_2 = \lambda##.

**How can I see that I have invariance under rotations about ##e_3##? Also, why does this imply that the two principal stretches are the same?**
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