Beams

A physically natural quadratic nonlinearity for thin elastic structures arises from **geometric nonlinearities** due to stretching of the midplane when the deflections are moderate but not infinitesimal. For a one-dimensional beam, the lowest-order nonlinear contribution to the bending strain can be expressed as the derivative of the squared slope:

$$\frac{\partial}{\partial x}\,(u_x^2) = (u_x^2)_x,$$

where $u(x,t)$ is the transverse displacement. This term is **quadratic in the amplitude** and represents the nonlinear coupling between curvature and slope, producing energy transfer between modes.

In two dimensions, for a thin plate or membrane $u(x,y,t)$, the corresponding geometric quadratic nonlinearity involves both spatial directions and can be written as

$$\frac{\partial}{\partial x}\big(u_x^2\big) + \frac{\partial}{\partial y}\big(u_y^2\big) = (u_x^2)_x + (u_y^2)_y,$$

where $u_x = \partial u / \partial x$ and $u_y = \partial u / \partial y$. Physically, this term captures the stretching of the midplane in both directions, generating **mode interactions in 2D** and allowing the transfer of energy among different linear eigenmodes of the plate. It is a minimal quadratic model that retains the essential physics of **triad resonances, amplitude-dependent frequency shifts, and nonlinear wave propagation** while remaining analytically tractable.

Introducing a physically natural quadratic nonlinearity of the form u u_xx + u u_yy, the full nonlinear plate equation reads:

$$\boxed{ u_{tt} + u_{xxxx} + u_{yyyy} + \varepsilon \left( u\, u_{... ...) = 0, \quad (x,y) \in \Omega \subset \mathbb{R}^2, \quad \varepsilon \ll 1, }$$

where &epsiv#varepsilon; is a small parameter controlling the strength of the nonlinearity.

$$H = \int \left[ \frac{1}{2}\,u_t^2 + \frac{1}{2}\left(u_{xx}^2 + ... ...2\right) - \frac{\varepsilon}{2}\,u\left(u_x^2 + u_y^2\right) \right]\;dx\,dy .$$ (12)

$$u_{\mathbf{k}} = \frac{1}{\sqrt{2\omega_{\mathbf{k}}}} \left( a_{\mathbf{k}} + a_{-\mathbf{k}}^{*} \right)$$ (9)

$$p_{\mathbf{k}} = - i \sqrt{\frac{\omega_{\mathbf{k}}}{2}} \left( a_{\mathbf{k}} - a_{-\mathbf{k}}^{*} \right)$$ (10)

$$H_0 = \sum_{\mathbf{k}} \omega_{\mathbf{k}} \, \vert a_{\mathbf{k}}\vert^2$$ (11)

$$H_1 = \frac{\varepsilon}{2} \sum_{\mathbf{k}_1+\mathbf{k}_2+\math... ...f{k}_3}^{*} + 3\,a_{\mathbf{k}_1} a_{\mathbf{k}_2} a_{\mathbf{k}_3}^{*} \right)$$ (12)

$$V_{\mathbf{k}_1\mathbf{k}_2\mathbf{k}_3} = \frac{ k_{2x} k_{3x} +... ...\sqrt{ 8\,\omega_{\mathbf{k}_1} \omega_{\mathbf{k}_2} \omega_{\mathbf{k}_3} } }$$ (13)

$$i \,\dot a_{\mathbf{k}} = \frac{\partial H}{\partial a_{\mathbf{k}}^{*}}$$ (14)

$$i \,\dot a_{\mathbf{k}} = \omega_{\mathbf{k}} a_{\mathbf{k}} + \v... ...{k}_1} a_{\mathbf{k}_2} \,\delta_{\mathbf{k}-\mathbf{k}_1-\mathbf{k}_2} +\cdots$$ (15)

$$u_{\mathbf{k}} = \frac{1}{\sqrt{2\omega_{\mathbf{k}}}} \left( a_{\mathbf{k}} + a_{-\mathbf{k}}^{*} \right)$$ (9)

$$p_{\mathbf{k}} = - i \sqrt{\frac{\omega_{\mathbf{k}}}{2}} \left( a_{\mathbf{k}} - a_{-\mathbf{k}}^{*} \right)$$ (10)

$$H_0 = \sum_{\mathbf{k}} \omega_{\mathbf{k}} \, \vert a_{\mathbf{k}}\vert^2$$ (11)

$$H_1 = \frac{\varepsilon}{2} \sum_{\mathbf{k}_1+\mathbf{k}_2+\math... ...f{k}_3}^{*} + 3\,a_{\mathbf{k}_1} a_{\mathbf{k}_2} a_{\mathbf{k}_3}^{*} \right)$$ (12)

$$V_{\mathbf{k}_1\mathbf{k}_2\mathbf{k}_3} = \frac{ k_{2x} k_{3x} +... ...\sqrt{ 8\,\omega_{\mathbf{k}_1} \omega_{\mathbf{k}_2} \omega_{\mathbf{k}_3} } }$$ (13)

$$i \,\dot a_{\mathbf{k}} = \frac{\partial H}{\partial a_{\mathbf{k}}^{*}}$$ (14)

$$i \,\dot a_{\mathbf{k}} = \omega_{\mathbf{k}} a_{\mathbf{k}} + \v... ...{k}_1} a_{\mathbf{k}_2} \,\delta_{\mathbf{k}-\mathbf{k}_1-\mathbf{k}_2} +\cdots$$ (15)