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The appearance of the second derivative (or Laplacian in higher dimensions) in the diffusion equation ($u_t=u_{xx}$) and the wave equation ($u_{tt}=u_{xx}$) seems intuitive to me. The quantity simply represents deviation from the local average - and the velocity or acceleration are proportional to this deviation.

On the other hand, the beam equation $u_{tt} = u_{xxxx}$ does not seem to conform to such intuition. While I understand the derivation of the equation, I still do not have any intuition about what the $u_{xxxx}$ term "actualy does".

The solutions to the beam equation behave similarly to the wave equation - can we think about $u_{xxxx}$ as a diffusion term in some way? How is it different to $u_{xx}$? I mean, what kind of dynamics do we get that is peculiar to the 4th order term?

In order to understand better how the 4th order term behaves differently to the 2nd order one, it might be useful to consider the equations $u_t=u_{xxxx}$ or $u_{tt}=u_{xx} + \varepsilon u_{xxxx}$ and compare with the standard diffusion/wave equation. But that is just an idea...

FusRoDah
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1 Answers1

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The fourth derivative arises because for the Euler–Bernoulli beam, the only deflection that matters is (very slight) curvature, and the only thing that produces curvature is the bending moment.

In other words, we're ignoring axial effects, large deflections that would alter the loading direction, and the impact of shear, among other implications.

Now, the bending moment needs to incorporate all the loads on the beam convolved with their corresponding distance. Let's say that a load $q(x)$ is distributed over the beam. We can integrate this distributed load as $\int_0^x q(x^\prime)dx^\prime$ to get the total force $F(x)$ applied from point $0$ to point $x$. We can integrate this force in turn as $\int_0^x F(x^{\prime})dx^{\prime}$ to get the total bending moment $M(x)$ at point $x$. Put another way, we could differentiate the bending moment twice to obtain the local distributed load: $M_{xx}(x)=q(x)$. This gives us two of the $x$ derivatives.

Now looking at the cross section at any point of the beam, a nonzero bending moment implies an uneven stress profile over that cross section, and this uneven stress induces curvature $\kappa$ in the beam, as mediated by the material stiffness $E$ of the beam and its geometric stiffness (the so-called area moment of inertia) $I$. For small curvature, we can approximate $\kappa$ by the second derivative of deflection, $w_{xx}(x)$, and so $EIw_{xx}(x)=M(x)$. This gives us the other two $x$ derivatives.

To this equation we differentiate twice and assume constant material and geometric properties to obtain $w_{xxxx}(x)\sim q(x)$. If the beam isn't at equilibrium, this force will induce acceleration: $w_{xxxx}\sim w_{tt}$. Is this the kind of interpretation you're looking for?

Chemomechanics
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