Review on dynamic mechanical analysis and design of graded cellular materials
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摘要: 多胞材料是一种内部含有大量空穴和胞元的结构，具有轻质、高比吸能等特性，广泛应用于航空航天、交通运输和人体防护等碰撞/爆炸防护领域。引入梯度设计可实现多胞材料的有序耗能和载荷调控，满足不同情形和工况下的防护需求。对梯度多胞材料动态力学行为和设计的研究进展进行了综述，着重介绍了梯度多胞材料/结构在抗冲击、抗爆炸和模拟爆炸载荷加载3个方面的应用案例。首先，介绍了梯度多胞材料的分类，较系统地总结了梯度多胞材料在动态加载下的变形特征、冲击波模型、防护性能等方面的研究，阐明了梯度多胞材料的密度/强度梯度与惯性效应存在的竞争机制。其次，以冲击波模型为力学原理指导梯度多胞材料/结构设计，介绍了梯度多胞材料耐撞性反向设计、多种抗爆炸夹芯结构设计、计及弹靶耦合效应的爆炸载荷模拟器设计等策略，实现了最佳防护效果或载荷精准控制，为抗冲击/抗爆炸防护设计和快速评估提供理论基础和技术支撑。最后，展望了梯度多胞材料在极端环境加载、大能量冲击和强非线性载荷调控等需求下冲击防护领域的应用前景。Abstract: Cellular materials are structures with a large number of internal cavities and cells, which have the properties of lightweight and high specific energy absorption, and they are widely used in the collision/explosion protection, such as aerospace, transportation, and human protection. Introducing a gradient design to cellular materials helps the materials to meet the protection requirements in different scenarios and conditions with the properties of orderly dissipation of energy and manipulation of loads. A review of research advances in the dynamic analysis and design on mechanical behavior of graded cellular materials is presented. Three cases of the applications of graded cellular materials/structures, i.e., impact resistance, blast resistance, and blast-mimicking loading, are elaborated. Firstly, graded cellular materials are briefly described from various aspects, such as natural vs. artificial, layered vs. continuous, strength gradient vs. density gradient, and conventional manufacturing vs. additive manufacturing. The studies of the deformation characteristics, shock wave models, and protective properties of graded cellular materials under dynamic loading are then reviewed systematically. A competitive mechanism of density/strength gradients and inertial effects exists in graded cellular materials to synergistically modulate collapse deformation modes. According to the stress-strain curve characteristics of cellular materials, choosing the appropriate constitutive model could increase the characterization accuracy for its dynamic mechanical behavior. Secondly, the shock wave models are used as a mechanical tool to guide the design of graded cellular materials/structures. Some strategies are elaborated, such as the backward design of graded cellular materials for impact resistance, the design of several types of anti-blast sandwich structures, and the design of blast-load simulators with the projectile-beam coupling effect being taken into account. The optimal protection effect or precise load control had been realized efficiently, which provides a theoretical basis and technical support for the protection design and rapid evaluation of impact/explosion resistance structures. Finally, for the applications in the scenarios of extreme environmental loading, large energy impacts, and strong nonlinear load manipulation, the investigations of graded cellular materials are full of challenges and expectation.
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表 1 不同冲击工况下梯度多胞材料的动态响应理论模型
Table 1. Theoretical models of the dynamic response for graded cellular materials under different impact scenarios
变形模式 质量块初速度撞击[80, 95] 爆炸加载[112] 单波 $ \left\{ \begin{gathered} \dot \varPhi = \dfrac{v}{{{\varepsilon _{\text{B}}}(\rho (\varPhi ))}} \\ {\sigma _{\text{B}}} = {\sigma _{\text{0}}} + {\rho _{\text{s}}}\rho (\varPhi )\dfrac{{{v^2}}}{{{\varepsilon _{\text{B}}}(\rho (\varPhi ))}} \\ \dot v = \dfrac{{ - {\sigma _{\text{B}}}}}{{m + {\rho _{\text{s}}}\displaystyle\int_0^\varPhi {\rho (X){\rm{d}}X} }} \\ \end{gathered} \right. $ $ \left\{ \begin{gathered} \dot \varPhi = \dfrac{v}{{{\varepsilon _{\text{B}}}(\rho (\varPhi ))}} \\ {\sigma _{\text{B}}} = {\sigma _{\text{0}}} + {\rho _{\text{s}}}\rho (\varPhi )\dfrac{{{v^2}}}{{{\varepsilon _{\text{B}}}(\rho (\varPhi ))}} \\ \dot v = \dfrac{{p(t) - {\sigma _{\text{B}}}}}{{m + {\rho _{\text{s}}}\displaystyle\int_0^\varPhi {\rho (X){\rm{d}}X} }} \\ \end{gathered} \right. $ 双波 $ \left\{ \begin{gathered} {{\dot \varPhi }_1} = \dfrac{{{v_1} - {v_2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _1}))}} \\ {{\dot \varPhi }_2} = \dfrac{{ - {v_2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _2}))}} \\ {\sigma _{{\text{B,1}}}} = {\sigma _{\text{0}}}(\rho ({\varPhi _1})) + {\rho _{\text{s}}}\rho ({\varPhi _1})\dfrac{{{{({v_1} - {v_2})}^2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _1}))}} \\ {\sigma _{{\text{B,2}}}} = {\sigma _{\text{0}}}(\rho ({\varPhi _2})) + {\rho _{\text{s}}}\rho ({\varPhi _2})\dfrac{{{v_2}^2}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _2}))}} \\ {{\dot v}_1} = \dfrac{{ - {\sigma _{{\text{B,1}}}}}}{{m + {\rho _{\text{s}}}\displaystyle\int_0^{{\varPhi _1}} {\rho (X){\text{d}}X} }} \\ {{\dot v}_2} = \dfrac{{{\sigma _{\text{0}}}(\rho ({\varPhi _1})) - {\sigma _{\text{0}}}(\rho ({\varPhi _2}))}}{{{\rho _{\text{s}}}\displaystyle\int_{{\varPhi _1}}^{{\varPhi _2}} {\rho (X){\text{d}}X} }} \\ \end{gathered} \right. $ $ \left\{ \begin{gathered} {{\dot \varPhi }_1} = \dfrac{{{v_1} - {v_2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _1}))}} \\ {{\dot \varPhi }_2} = \dfrac{{ - {v_2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _2}))}} \\ {\sigma _{{\text{B,1}}}} = {\sigma _{\text{0}}}(\rho ({\varPhi _1})) + {\rho _{\text{s}}}\rho ({\varPhi _1})\dfrac{{{{({v_1} - {v_2})}^2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _1}))}} \\ {\sigma _{{\text{B,2}}}} = {\sigma _{\text{0}}}(\rho ({\varPhi _2})) + {\rho _{\text{s}}}\rho ({\varPhi _2})\dfrac{{{v_2}^2}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _2}))}} \\ {{\dot v}_1} = \dfrac{{p(t) - {\sigma _{{\text{B,1}}}}}}{{m + {\rho _{\text{s}}}\displaystyle\int_0^{{\varPhi _1}} {\rho (X){\text{d}}X} }} \\ {{\dot v}_2} = \dfrac{{{\sigma _{\text{0}}}(\rho ({\varPhi _1})) - {\sigma _{\text{0}}}(\rho ({\varPhi _2}))}}{{{\rho _{\text{s}}}\displaystyle\int_{{\varPhi _1}}^{{\varPhi _2}} {\rho (X){\text{d}}X} }} \\ \end{gathered} \right. $ 三波 $ \left\{ \begin{gathered} {{\dot \varPhi }_1} = \dfrac{{{v_1} - {v_2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _1}))}} \\ {{\dot \varPhi }_2} = \dfrac{{{v_2} - {v_3}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _2}))}} \\ {{\dot \varPhi }_3} = \dfrac{{{v_3}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _3}))}} \\ {\sigma _{{\text{B,1}}}} = {\sigma _{\text{0}}}(\rho ({\varPhi _1})) + {\rho _{\text{s}}}\rho ({\varPhi _1})\dfrac{{{{({v_1} - {v_2})}^2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _1}))}} \\ {\sigma _{{\text{B,2}}}} = {\sigma _{\text{0}}}(\rho ({\varPhi _2})) + {\rho _{\text{s}}}\rho ({\varPhi _2})\dfrac{{{{({v_2} - {v_3})}^2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _2}))}} \\ {\sigma _{{\text{B,3}}}} = {\sigma _{\text{0}}}(\rho ({\varPhi _3})) + {\rho _{\text{s}}}\rho ({\varPhi _3})\dfrac{{{v_3}^2}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _3}))}} \\ {{\dot v}_1} = \dfrac{{ - {\sigma _{{\text{B,1}}}}(\rho ({\varPhi _1}))}}{{m + {\rho _{\text{s}}}\displaystyle\int_0^{{\varPhi _1}} {\rho (X){\rm{d}}X} }} \\ {{\dot v}_2} = \dfrac{{{\sigma _{\text{0}}}(\rho ({\varPhi _1})) - {\sigma _{\text{0}}}(\rho ({\varPhi _2}))}}{{{\rho _{\text{s}}}\displaystyle\int_{{\varPhi _1}}^{{\varPhi _2}} {\rho (X){\rm{d}}X} }} \\ {{\dot v}_3} = \dfrac{{{\sigma _{{\text{B,3}}}}(\rho ({\varPhi _2})) - {\sigma _{{\text{B,2}}}}(\rho ({\varPhi _3}))}}{{{\rho _{\text{s}}}\displaystyle\int_{{\varPhi _2}}^{{\varPhi _3}} {\rho (X){\rm{d}}X} }} \\ \end{gathered} \right. $ $ \left\{ \begin{gathered} {{\dot \varPhi }_1} = \dfrac{{{v_1} - {v_2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _1}))}} \\ {{\dot \varPhi }_2} = \dfrac{{{v_2} - {v_3}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _2}))}} \\ {{\dot \varPhi }_3} = \dfrac{{{v_3}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _3}))}} \\ {\sigma _{{\text{B,1}}}} = {\sigma _{\text{0}}}(\rho ({\varPhi _1})) + {\rho _{\text{s}}}\rho ({\varPhi _1})\dfrac{{{{({v_1} - {v_2})}^2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _1}))}} \\ {\sigma _{{\text{B,2}}}} = {\sigma _{\text{0}}}(\rho ({\varPhi _2})) + {\rho _{\text{s}}}\rho ({\varPhi _2})\dfrac{{{{({v_2} - {v_3})}^2}}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _2}))}} \\ {\sigma _{{\text{B,3}}}} = {\sigma _{\text{0}}}(\rho ({\varPhi _3})) + {\rho _{\text{s}}}\rho ({\varPhi _3})\dfrac{{{v_3}^2}}{{{\varepsilon _{\text{B}}}(\rho ({\varPhi _3}))}} \\ {{\dot v}_1} = \dfrac{{p(t) - {\sigma _{{\text{B,1}}}}(\rho ({\varPhi _1}))}}{{m + {\rho _{\text{s}}}\displaystyle\int_0^{{\varPhi _1}} {\rho (X){\rm{d}}X} }} \\ {{\dot v}_2} = \dfrac{{{\sigma _{\text{0}}}(\rho ({\varPhi _1})) - {\sigma _{\text{0}}}(\rho ({\varPhi _2}))}}{{{\rho _{\text{s}}}\displaystyle\int_{{\varPhi _1}}^{{\varPhi _2}} {\rho (X){\rm{d}}X} }} \\ {{\dot v}_3} = \dfrac{{{\sigma _{{\text{B,3}}}}(\rho ({\varPhi _2})) - {\sigma _{{\text{B,2}}}}(\rho ({\varPhi _3}))}}{{{\rho _{\text{s}}}\displaystyle\int_{{\varPhi _2}}^{{\varPhi _3}} {\rho (X){\rm{d}}X} }} \\ \end{gathered} \right. $ 
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