Error estimation in the Bernoulli–Navier model for elastic rods

TL;DRAbstract

Let $u(\varepsilon)$ be the rescaled three‐dimensional displacement field solution of the linear elastic model for a clamped prismatic straight rod ${\varOmega}^\varepsilon$ having cross section with diameter of order $\varepsilon$ , and let $u^0$ be the corresponding Bernoulli–Navier displacement. In this article we establish that the error $\|u(\varepsilon)-u^0\|_{1,{\varOmega}}$ in the reference space $[H^1({\varOmega})]^3$ is of order $\varepsilon^{1/2}$ . We mainly use an auxiliar corrector function and we prove that this estimation cannot be improved using other corrector functions of the same family.

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