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Mechanics_student

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- TL;DR Summary
- Rigid body motion (RBM) transformation of interface points between a rigid and an elastic bodies in contact

Above are pictures of a problem in mechanics. An elastic body occupying domain ##\Omega## is supported below with a fixed support and from above with a rigid body. The following calculations aim to express the movements of the points located at the interface ##\Gamma_F## between the rigid and elastic bodies in terms of a reference point ##B##:

I don't understand them well, so I will explain my understanding (not sure it's correct), and confusion.

$$\vec{y}(x) = \vec{x} + \vec{u}(x)$$

The new position ##\vec{y}## of a point is equal to its original position ##\vec{x}## plus its displacement ##\vec{u}##.

Rigid body motion transformation:

$$\vec{y}(\vec{x}) = \vec{y}^B + \mathcal{R}(\beta) (\vec{x} - \vec{x}^B)$, where $\vec{x} \in \Gamma_F$$

where ##\mathcal{R}## is the rotation matrix as the rigid body rotates at an angle ##\beta##.

$$\mathcal{R} = \begin{bmatrix}

cos(\beta) & sin(\beta)\\

-sin(\beta) & cos(\beta)

\end{bmatrix}$$

Linearization for small deformation:

$$\vec{u}(x) - \vec{u}^B = \vec{y} - \vec{y}^B - (\vec{x} - \vec{x}^B)\\

= (\mathcal{R}(\beta) - I)(\vec{x}-\vec{x}^B)$$

where ##I## is the identity matrix.

For linearization we take,

$$\mathcal{R} = \begin{bmatrix}

0 & \beta\\

-\beta & 0

\end{bmatrix}$$

$$\Rightarrow \vec{u}(x)\mid_{\Gamma_F} = \vec{u}^B + (\mathcal{R}(\beta) - I)(\vec{x}-\vec{x}^B)$$

I don't understand the displacement of the points on the interface are parametrize by the displacement of a reference point that is also located on the interface undergoing transformation as well.

Isn't a reference point supposed to be fixed?

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