Lesson plan
Flamant solution
In continuum mechanics, the Flamant solution provides expressions for the stresses and displacements in a linear elastic wedge loaded by point forces at its sharp end. This solution was developed by in 1892 by modifying the three dimensional solutions for linear elasticity of Joseph Valentin Boussinesq. The stresses predicted by the Flamant solution are (in polar coordinates) : \begin{align} \sigma_{rr} & = \frac{2C_1\cos\theta}{r} + \frac{2C_3\sin\theta}{r} \\ \sigma_{r\theta} & = 0 \\ \sigma_{\theta\theta} & = 0 \end{align} where C_1, C_3 are constants that are determined from the boundary conditions and the geometry of the wedge (i.e., the angles \alpha,\beta ) and satisfy : \begin{align} F_1 & + 2\int_{\alpha}^{\beta} (C_1\cos\theta + C_3\sin\theta)\,\cos\theta\, d\theta = 0 \\ F_2 & + 2\int_{\alpha}^{\beta} (C_1\cos\theta + C_3\sin\theta)\,\sin\theta\, d\theta = 0 \end{align} where F_1,F_2 are the applied forces. The wedge problem is self-similar and has no inherent length scale. Also, all quantities can be expressed in the separated-variable form \sigma = f(r)g(\theta) . The stresses vary as (1/r) .
Article summary
In continuum mechanics, the Flamant solution provides expressions for the stresses and displacements in a linear elastic wedge loaded by point forces at its sharp end. This solution was developed by in 1892 by modifying the three dimensional solutions for linear elasticity of Joseph Valentin Boussinesq. The stresses predicted by the Flamant solution are (in polar coordinates) : \begin{align} \sigma_{rr} & = \frac{2C_1\cos\theta}{r} + \frac{2C_3\sin\theta}{r} \\ \sigma_{r\theta} & = 0 \\ \sigma_{\theta\theta} & = 0 \end{align} where C_1, C_3 are constants that are determined from the boundary conditions and the geometry of the wedge (i.e., the angles \alpha,\beta ) and satisfy : \begin{align} F_1 & + 2\int_{\alpha}^{\beta} (C_1\cos\theta + C_3\sin\theta)\,\cos\theta\, d\theta = 0 \\ F_2 & + 2\int_{\alpha}^{\beta} (C_1\cos\theta + C_3\sin\theta)\,\sin\theta\, d\theta = 0 \end{align} where F_1,F_2 are the applied forces. The wedge problem is self-similar and has no inherent length scale. Also, all quantities can be expressed in the separated-variable form \sigma = f(r)g(\theta) . The stresses vary as (1/r) .
Key claims
No claims recorded.
Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
No sources recorded.
Key claims and bibliography are generated from the same tracked claims and sources as the article page.