预应力下石墨烯增强复合材料板壳结构的非线性动力学行为

Due to its low mass density, superior mechanical, thermal and electrical properties, graphene can be widely used in sensors, energy storage devices, healthcare and as a coating against corrosion since its discovery in 2004. In actual working conditions, the mechanical behavior of many graphene-reinforced composite plate and shell structures can be described by dynamical problems in pre-stressed states. Therefore, it is very important to understand the dynamic behavior of graphene-reinforced composite plate and shell structures under pre-stresses. In this project, due to hyperelastic constitutive models, based on the finite deformation theory, theory of small deformations superposed on large elastic deformations and modern theory of nonlinear dynamics, through mathematical modelling, theoretical analysis, analytical/numerical solution and numerical simulation, the nonlinear dynamics behavior of pre-stressed spherical shells, cylindrical shells and plates composed of graphene-reinforced composites under radial dynamic loads will be investigated synthetically. The approximately analytical and numerical approaches will be developed for nonlinear vibration of plates and shells composed of graphene-reinforced composites. The effects of related parameters, especially the pre-stresses, on the nonlinear dynamic behavior of vibration modes, bifurcations and chaos are discussed. The research results of this project will enrich the nonlinear vibration theory of the continuous system, clarify the influence of pre-stresses on the nonlinear vibration system, and provide a theoretical basis for the optimization and design of graphene-reinforced composite plate and shell structures under pre-stresses.

石墨烯，自2004年被发现以来，因其具有低质量密度，良好的力学，导热，以及导电性能，在传感器，能量存储装置，医疗服务，以及防腐涂层等领域具有广泛应用。在实际工况中，很多石墨烯增强复合材料板壳结构的力学行为都可以用预应力下的动力学模型来描述。因此，认知预应力下石墨烯增强复合材料板壳结构的非线性动力学行为非常必要。本项目借助超弹性本构模型，运用有限变形理论、大变形叠加小变形理论和非线性动力学理论，通过数学建模、理论分析、解析/数值求解和数值模拟，综合研究预应力下由石墨烯增强复合材料组成的球壳，圆柱壳和板受径向动载荷时的非线性动力学行为。将发展针对该非线性振动分析的近似解析方法和数值方法，讨论相关参数，特别是预应力，对振动模态，分岔以及混沌等非线性动力学行为的影响。研究成果将丰富连续系统的非线性振动理论，明确预应力对系统非线性振动的影响，为优化和设计石墨烯增强复合材料板壳结构提供理论基础。

在研究石墨烯增强复合材料板壳结构的非线性振动行为时，可以利用超弹性模型描述材料的本构关系。本项目的主要研究内容如下：（1）研究了不可压缩neo-Hookean材料圆柱壳在翻转预应力状态下的稳定性问题。首先，利用大变形叠加小变形的思想，建立了描述圆柱壳翻转后轴向受压的稳定性的数学模型。其次，在翻转预应力作用下，利用增量方程理论，推导出描述翻转后受到轴向载荷作用下的圆柱壳失稳的增量方程和增量边界条件，进而得到了增量方程的通解以及失稳临界状态时描述圆柱壳有限变形的特解。最后，通过数值模拟，得到了圆柱壳初始厚度和粗细比对临界控制参数轴向伸长率的影响，并给出了失稳临界态圆柱壳内、外表面的有限变形。（2）研究了由可压缩neo-Hookean材料组成的圆柱壳的稳态波的径向传播问题。首先，建立了描述该圆柱壳受到径向拉伸后的有限变形的数学模型，利用多重打靶法求出问题的数值解。其次，根据大变形叠加小变形理论，将一个动态小扰动叠加到有限变形后的具有预应力的圆柱壳上，得到了稳态波运动的存在性。最后，利用数值模拟，讨论了径向静载荷和材料参数，以及内外半径比对稳态波的影响。（3）研究了由不可压缩Mooney-Rivlin材料组成的薄壁圆柱短壳在径向简谐激励作用下非线性振动问题，展示了在超弹性材料组成的圆柱壳中存在2:1内共振现象。首先，基于Kirchhoff-love假设，Donnell非线性扁壳理论以及能量变分原理，得到了描述圆柱壳径向运动的耦合微分方程。其次，通过分析不同模态的固有频率给出了2:1内共振的存在条件。最后，基于多尺度法得到了幅频响应关系，确定了稳态响应的稳定性。