Nonlinear Dynamic Response of an Abruptly Loaded Rubber-Like Hyperelastic Plate Resting on a Dissipative Viscoelastic Winkler–Pasternak Medium

Abstract

Sometimes, the engineering layers and living tissues may be modeled as hyperelastic plates resting on dissipative viscoelastic foundations featuring shear tractions at the contact surfaces. In this paper, the damped forced vibration and lateral deflections of thin neo-Hookean hyperelastic plates with various edge supports and Winkler–Pasternak viscoelastic foundations are studied under step and harmonic dynamic distributed loads. Using Hamilton’s principle, von Kármán’s assumptions, and classical plate theory, the governing equations of motion are obtained. Then, the combination of the motion equations and boundary conditions is solved by the iterative 2D differential quadrature and Newmark methods. The influence of the thickness and aspect ratio of the plate, shear stiffness and damping parameters of the viscoelastic foundation, type of edge conditions, the constitutive parameters of the hyperelastic material, the magnitude of the distributed load, and the excitation frequency on the resulting dynamic lateral deflections, dissipation manner of the vibration, and performance of the structure are studied. Furthermore, results show that the linearization errors are significant for larger loads and looser boundary conditions.

Appendix

Substituting Eqs. (15), and (18) to (20) into Eq. (21) and setting the expressions multiplied by $\delta u\text{,} \delta v,$ and $\delta w$ equal to zero, the following motion equations may be obtained:

$$4h{\mu }_{1}\frac{{\partial }^{2}{u}_{0}\left(X,Y,t\right)}{\partial {X}^{2}}+\frac{1}{4}h{\mu }_{1}\frac{{\partial }^{2}{u}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}}+\frac{9}{4}h{\mu }_{1}\frac{{\partial }^{2}{v}_{0}\left(X,Y,t\right)}{\partial Y\partial X}+\frac{9}{4}h{\mu }_{1}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial Y\partial X}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial Y}+\frac{1}{4}h{\mu }_{1}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial X}+4h{\mu }_{1}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial X}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {X}^{2}}-\rho h\frac{{\partial }^{2}{u}_{0}\left(X,Y,t\right)}{\partial {t}^{2}}=0$$

$$\frac{9}{4}h{\mu }_{1}\frac{{\partial }^{2}{u}_{0}\left(X,Y,t\right)}{\partial Y\partial X}+4h{\mu }_{1}\frac{{\partial }^{2}{v}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}}+\frac{1}{4}h{\mu }_{1}\frac{{\partial }^{2}{v}_{0}\left(X,Y,t\right)}{\partial {X}^{2}}+\frac{1}{4}h{\mu }_{1}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {X}^{2}}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial Y}+\frac{9}{4}h{\mu }_{1}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial Y\partial X}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial X}+4h{\mu }_{1}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial Y}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}}-\rho h\frac{{\partial }^{2}{v}_{0}\left(X,Y,t\right)}{\partial {t}^{2}}=0$$

$$-\frac{1}{3}{h}^{3}{\mu }_{1}\frac{{\partial }^{4}{w}_{0}\left(X,Y,t\right)}{\partial {Y}^{4}}-\frac{5}{12}{h}^{3}{\mu }_{1}\frac{{\partial }^{4}{w}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}\partial {X}^{2}}-\frac{1}{3}{h}^{3}{\mu }_{1}\frac{{\partial }^{4}{w}_{0}\left(X,Y,t\right)}{\partial {X}^{4}}+5h{\mu }_{1}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial X\partial Y}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial X}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial Y}+\frac{9}{4}h{\mu }_{1}\left[\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial X}\frac{{\partial }^{2}{v}_{0}\left(X,Y,t\right)}{\partial X\partial Y}+\frac{{\partial }^{2}{u}_{0}\left(X,Y,t\right)}{\partial Y\partial X}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial Y}\right]+\frac{1}{2}h{\mu }_{1}\left[\frac{\partial {v}_{0}\left(X,Y,t\right)}{\partial X}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial X\partial Y}+\frac{\partial {u}_{0}\left(X,Y,t\right)}{\partial Y}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial Y\partial X}\right]+\frac{1}{4}h{\mu }_{1}\left[\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial X}\frac{{\partial }^{2}{u}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}}+\frac{{\partial }^{2}{v}_{0}\left(X,Y,t\right)}{\partial {X}^{2}}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial Y}\right]+4h{\mu }_{1}\left[\frac{{\partial }^{2}{v}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial Y}+\frac{\partial {v}_{0}\left(X,Y,t\right)}{\partial Y}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}}\right]+2h{\mu }_{1}\frac{\partial {u}_{0}\left(X,Y,t\right)}{\partial X}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}}+4h{\mu }_{1}\frac{{\partial }^{2}{u}_{0}\left(X,Y,t\right)}{\partial {X}^{2}}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial X}+4h{\mu }_{1}\frac{\partial {u}_{0}\left(X,Y,t\right)}{\partial X}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {X}^{2}}+2h{\mu }_{1}\frac{\partial {v}_{0}\left(X,Y,t\right)}{\partial Y}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {X}^{2}}+\frac{5}{4}h{\mu }_{1}\left[{\left(\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial Y}\right)}^{2}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {X}^{2}}+{\left(\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial X}\right)}^{2}\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}}\right]+6h{\mu }_{1}\left[\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {X}^{2}}{\left(\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial X}\right)}^{2}+\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}}{\left(\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial Y}\right)}^{2}\right]+{K}_{g}\left[\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {X}^{2}}+\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}}\right]-{K}_{w}{w}_{0}\left(X,Y,t\right)+{C}_{d}\frac{\partial {w}_{0}\left(X,Y,t\right)}{\partial t}-\rho h\frac{{\partial }^{2}{w}_{0}\left(X,Y,t\right)}{\partial {t}^{2}}+\frac{1}{12}\rho {h}^{3}\left[\frac{{\partial }^{4}{w}_{0}\left(X,Y,t\right)}{\partial {X}^{2}\partial {t}^{2}}+\frac{{\partial }^{4}{w}_{0}\left(X,Y,t\right)}{\partial {Y}^{2}\partial {t}^{2}}\right]=F\left(X,Y,t\right)$$