A Note on Elastic Deformation

math

pde

Nguyen Minh Hieu

12/07/2023

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\begin{aligned} \mathbf{u}(\mathbf{x}, t) = \begin{bmatrix} u_1(\mathbf{x},t) \\ u_2(\mathbf{x},t) \\ u_3(\mathbf{x},t) \end{bmatrix} \end{aligned}

Stress denotes a force on surface $i$ in the direction $j$

\begin{aligned} \sigma_{ij} = \bm{\sigma}(\mathbf{x}, t) = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix} \end{aligned}

Strain denotes the variation of the displacement under stress

\begin{aligned} u_i(\mathbf{x} + \delta\mathbf{x}) \approx u_i(\mathbf{x}) + \frac{\partial u_i(\mathbf{x})}{\partial x_j} \delta x_j = \underbrace{u_i(\mathbf{x})}_{translation} + \underbrace{\delta u_i(\mathbf{x})}_{rotation + deformation} \end{aligned}

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