局部应变场计算方法及其在多胞材料和结构中的应用
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摘要
多胞材料具有复杂的细观结构,存在明显的多尺度特征。由于细观结构的失稳而导致变形局部化是多胞材料准静态及动态响应的典型特征。基于连续介质力学定义的名义应变无法表征这种局部化的变形,而现有的局部应变计算方法尚有不同程度的局限性。本文的目的就是发展一种新的局部应变场计算方法以较好地表征多胞材料的局部化变形,并利用该方法研究多胞材料以及多胞材料复合结构的动态力学行为。
本文基于离散局部变形梯度发展了局部应变场计算方法以表征多胞材料的非均匀变形。多胞材料首先被离散化为一系列节点,对于某一给定节点,由该节点及其周围一定范围内其他节点的相对运动计算出该节点位置处的基于最小二乘误差的最佳局部变形梯度。然后基于连续介质理论由局部变形梯度构造局部应变张量。本文发展的局部应变场计算方法可以考虑有限变形状态,既适合于规则多胞材料,也适合于不规则多胞材料。
借助局部应变场计算方法,我们研究了塑性冲击波在多胞材料中的传播。Voronoi蜂窝二维应变场显示高速和中等速度冲击下均存在一维塑性冲击波。基于塑性冲击波传播的一维性,由二维应变场得到了沿加载方向的一维应变分布。应变分布揭示多胞材料中传播的冲击波,其波阵面处同样具有不连续性,就如连续固体中传播的冲击波一样,为一维冲击波模型的应用提供了基础。基于应变分布曲线,我们提出了确定冲击波波阵面位置的方法,由此进一步得到了冲击波的速度。这些基于胞元的细观有限元模型结果校验了基于连续介质力学的一维冲击波模型,发现基于非线性应变硬化材料模型的冲击波模型能够更准确地预测冲击波的速度。最后,我们还获取了冲击波波阵面两侧的应力-应变状态,从变形机理上解释了多胞材料动态压溃实验和数值模拟中均观察到的支撑端应力随冲击速度增大而减小的有趣现象。
在我们发展的局部应变场计算方法中,截断半径决定了参与构建给定节点处最佳局部变形梯度的周围节点的个数,因此,局部应变场方法的局部性和准确性依赖于截断半径。本文对截断半径的敏感性进行了分析,进一步完善了局部应变场计算方法。我们引入了两种截断半径的可选方案,以表征恒速压溃下Voronoi蜂窝的非均匀变形为例对两种方案中截断半径的取值给出了建议,并进一步提出了截断半径的一种优化方案。计算所得平均应变与名义应变之间的比较结果表明,不论蜂窝是低速还是高速压溃,局部应变场计算方法的截断半径的优化方案均可以较好地表征多胞材料细观尺度上非均匀的变形。最后,利用局部应变场计算方法的截断半径的优化方案表征了两种不同构型的双层多胞牺牲层(DLCC)在三角脉冲爆炸载荷作用下的变形。二维应变场和一维应变分布揭示了塑性冲击波在两种构型中的传播规律,为双层多胞牺牲层的设计提供了建设性的思路。
本文进一步提出了抗爆炸双层多胞牺牲层的设计方法。考察了两种不同构型的双层多胞牺牲层：一种是两层多胞材料芯层具有相同密度的构型,即Cladding-1,另一种是两层多胞材料芯层密度不同且高密度多胞材料靠近爆炸载荷、低密度多胞材料靠近支撑端的构型,即Cladding-2。利用经典的一维冲击波模型研究了两种构型双层多胞牺牲层中塑性冲击波的传播,发现Cladding-1中只有单一冲击波而Cladding-2中有两个冲击波同时传播。理论推导了Cladding-1完全吸收爆炸载荷所需的最小厚度即临界厚度的显式表达式。分析了Cladding-2的响应特征,在此基础上引入两个约束条件优化了Cladding-2中近层和远层芯层厚度的配置并得到了Cladding-2的临界厚度。在面板总质量相等的前提下,比较了单层多胞牺牲层(SLCC)和两种双层多胞牺牲层Cladding-1、Cladding-2的临界厚度,发现SLCC总是比Cladding-1更高效,而Cladding-2则可以比SLCC更高效。而且,通过选择最优的面板质量比和芯层密度比可以使Cladding-2的临界厚度最小。然后,我们提出了一种设计方法指导双层多胞牺牲层的抗爆炸防护设计。最后,利用Cladding-2的基于胞元的细观有限元模型校验了基于连续介质力学的一维冲击波模型的理论分析结果,两者整体吻合较好。借助局部应变场计算方法所得应变分布对导致理论预测结果和细观有限元模型结果出现偏差的原因进行了分析。
Cellular materials have complex meso-structrue and apparent multiscale features. Deformation localization caused by the collapse of meso-structures is a typical feature of both quasi-static and dynamic responses of cellular materials. The nominal strain, defined based on the continuum mechanics, can not characterize the localized deformation, and methods for calculating the local strain proposed in the literature have some limitations. This thesis aims at developing a new method to characterize the localized deformation in the cellular materials and further investigate the dynamic responses of cellular materials and cellular-core composite structures.
Local strain field calculation method based on discrete local deformation gradient is developed to characterize the heterogeneous deformation of cellular materials. In this approach, cellular materials are discretized into a series of nodes and the optimal local deformation gradient at a representative node, in a least squares sense, is calculated by the relative motions of the representative node and its neighboring nodes. Then, a local strain tensor related to the optimal local deformation gradient can be calculated based on the continuum mechanics theory. The local strain field calculation method allows considering general finite-strain states of both regular and irregular cellular materials.
Plastic shock wave propagation in a cellular material is investigated by using the local strain field calculation method.2D strain fields of Voronoi honeycomb specimens provide evidences of the existence of a one-dimensional plastic shock front in the cellular material under a high or moderate velocity impact. Due to the feature of the shock front propagation, the2D strain fields are simplified to one-dimensional strain distributions in the loading direction. The strain distributions reveal that there exists a zone at the shock front across which there is a discontinuity in strain, which is just like a shock front in a solid continuum. This enhances the basis of one-dimensional shock models. Shock wave velocity is measured by an approach that gives the location of the shock front varying with the impact time. A comparison of the cell-based finite element model with continuum-based shock models indicates that the shock model based on a material accounting for the post-locking behaviour is much accurate in predicting the shock wave velocity. Finally, stress-strain states ahead of and behind the shock front are obtained. These results provide an explanation in terms of deformation mechanism for the stress reduction at the support end with increasing impact velocity, which was previously observed in experimental and numerical studies.
In the local strain field calculation method, the cut-off radius determines the neighboring nodes of a representative node for calculating the optimal local deformation gradient. Therefore, the local nature and accuracy of the method are dependent on the cut-off radius. The sensitivity of the cut-off radius is investigated to further improve the local strain field calculation method. Two different schemes are first analyzed to determine the suitable cut-off radii by characterizing the heterogeneous deformation of Voronoi honeycombs under uniaxial compression. Then, an optimal scheme is further suggested. It is demonstrated that the local strain field calculation method with the optimal scheme characterizes the heterogeneous deformation of cellular materials reasonable well whether the compression rate is low or high. Finally, the local strain field calculation method is applied to reveal the evolution of the heterogeneous deformation of two different configurations of double-layer cellular claddings (DLCCs) under a linear decaying blast load.2D fields and ID distributions of local engineering strain are calculated. These results interpret the shock wave propagation mechanisms in both claddings and provide a useful understanding in the design of a double-layer cellular cladding.
This thesis further presents a method for the design of a double-layer cellular cladding serving as a protective function against blast load. Two configurations of DLCCs are considered, i.e. cladding with cellular cores of identical density (Cladding-1) and cladding with a higher density cellular core layer close to the blast load (Cladding-2). Shock wave propagation in the two DLCCs is investigated by using the classical one-dimensional shock model. Single shock front propagates in Cladding-1, while double shock fronts propagate in Cladding-2. A closed-form solution of the critical thickness, which is the minimum thickness required to fully absorb the blast load, is given for Cladding-1. Response features of Cladding-2are analyzed and then the critical thickness of Cladding-2is determined by optimizing the layer thicknesses through introducing two constraints. With an equal total mass of cover plates, Cladding-1and Cladding-2are compared with a single-layer cellular cladding (SLCC) in terms of the critical thickness. It is demonstrated that the SLCC is always more efficient than Cladding-1as a protective structure for blast alleviation, while Cladding-2can be more efficient than the SLCC. Moreover, a maximum reduction of the critical thickness of Cladding-2can be achieved by choosing an optimal combination of density ratio of the two cellular cores and mass ratio of the two cover plates. A design method for Cladding-2against blast load is further presented. Finally, cell-based finite element simulations are carried out to verify the analytical predictions about the design of Cladding-2. Comparisons between the cell-based finite element results and the analytical predictions are generally good, and some difference is analyzed through the strain distributions measured by the local strain field calculation method.
本文基于离散局部变形梯度发展了局部应变场计算方法以表征多胞材料的非均匀变形。多胞材料首先被离散化为一系列节点,对于某一给定节点,由该节点及其周围一定范围内其他节点的相对运动计算出该节点位置处的基于最小二乘误差的最佳局部变形梯度。然后基于连续介质理论由局部变形梯度构造局部应变张量。本文发展的局部应变场计算方法可以考虑有限变形状态,既适合于规则多胞材料,也适合于不规则多胞材料。
借助局部应变场计算方法,我们研究了塑性冲击波在多胞材料中的传播。Voronoi蜂窝二维应变场显示高速和中等速度冲击下均存在一维塑性冲击波。基于塑性冲击波传播的一维性,由二维应变场得到了沿加载方向的一维应变分布。应变分布揭示多胞材料中传播的冲击波,其波阵面处同样具有不连续性,就如连续固体中传播的冲击波一样,为一维冲击波模型的应用提供了基础。基于应变分布曲线,我们提出了确定冲击波波阵面位置的方法,由此进一步得到了冲击波的速度。这些基于胞元的细观有限元模型结果校验了基于连续介质力学的一维冲击波模型,发现基于非线性应变硬化材料模型的冲击波模型能够更准确地预测冲击波的速度。最后,我们还获取了冲击波波阵面两侧的应力-应变状态,从变形机理上解释了多胞材料动态压溃实验和数值模拟中均观察到的支撑端应力随冲击速度增大而减小的有趣现象。
在我们发展的局部应变场计算方法中,截断半径决定了参与构建给定节点处最佳局部变形梯度的周围节点的个数,因此,局部应变场方法的局部性和准确性依赖于截断半径。本文对截断半径的敏感性进行了分析,进一步完善了局部应变场计算方法。我们引入了两种截断半径的可选方案,以表征恒速压溃下Voronoi蜂窝的非均匀变形为例对两种方案中截断半径的取值给出了建议,并进一步提出了截断半径的一种优化方案。计算所得平均应变与名义应变之间的比较结果表明,不论蜂窝是低速还是高速压溃,局部应变场计算方法的截断半径的优化方案均可以较好地表征多胞材料细观尺度上非均匀的变形。最后,利用局部应变场计算方法的截断半径的优化方案表征了两种不同构型的双层多胞牺牲层(DLCC)在三角脉冲爆炸载荷作用下的变形。二维应变场和一维应变分布揭示了塑性冲击波在两种构型中的传播规律,为双层多胞牺牲层的设计提供了建设性的思路。
本文进一步提出了抗爆炸双层多胞牺牲层的设计方法。考察了两种不同构型的双层多胞牺牲层：一种是两层多胞材料芯层具有相同密度的构型,即Cladding-1,另一种是两层多胞材料芯层密度不同且高密度多胞材料靠近爆炸载荷、低密度多胞材料靠近支撑端的构型,即Cladding-2。利用经典的一维冲击波模型研究了两种构型双层多胞牺牲层中塑性冲击波的传播,发现Cladding-1中只有单一冲击波而Cladding-2中有两个冲击波同时传播。理论推导了Cladding-1完全吸收爆炸载荷所需的最小厚度即临界厚度的显式表达式。分析了Cladding-2的响应特征,在此基础上引入两个约束条件优化了Cladding-2中近层和远层芯层厚度的配置并得到了Cladding-2的临界厚度。在面板总质量相等的前提下,比较了单层多胞牺牲层(SLCC)和两种双层多胞牺牲层Cladding-1、Cladding-2的临界厚度,发现SLCC总是比Cladding-1更高效,而Cladding-2则可以比SLCC更高效。而且,通过选择最优的面板质量比和芯层密度比可以使Cladding-2的临界厚度最小。然后,我们提出了一种设计方法指导双层多胞牺牲层的抗爆炸防护设计。最后,利用Cladding-2的基于胞元的细观有限元模型校验了基于连续介质力学的一维冲击波模型的理论分析结果,两者整体吻合较好。借助局部应变场计算方法所得应变分布对导致理论预测结果和细观有限元模型结果出现偏差的原因进行了分析。
Cellular materials have complex meso-structrue and apparent multiscale features. Deformation localization caused by the collapse of meso-structures is a typical feature of both quasi-static and dynamic responses of cellular materials. The nominal strain, defined based on the continuum mechanics, can not characterize the localized deformation, and methods for calculating the local strain proposed in the literature have some limitations. This thesis aims at developing a new method to characterize the localized deformation in the cellular materials and further investigate the dynamic responses of cellular materials and cellular-core composite structures.
Local strain field calculation method based on discrete local deformation gradient is developed to characterize the heterogeneous deformation of cellular materials. In this approach, cellular materials are discretized into a series of nodes and the optimal local deformation gradient at a representative node, in a least squares sense, is calculated by the relative motions of the representative node and its neighboring nodes. Then, a local strain tensor related to the optimal local deformation gradient can be calculated based on the continuum mechanics theory. The local strain field calculation method allows considering general finite-strain states of both regular and irregular cellular materials.
Plastic shock wave propagation in a cellular material is investigated by using the local strain field calculation method.2D strain fields of Voronoi honeycomb specimens provide evidences of the existence of a one-dimensional plastic shock front in the cellular material under a high or moderate velocity impact. Due to the feature of the shock front propagation, the2D strain fields are simplified to one-dimensional strain distributions in the loading direction. The strain distributions reveal that there exists a zone at the shock front across which there is a discontinuity in strain, which is just like a shock front in a solid continuum. This enhances the basis of one-dimensional shock models. Shock wave velocity is measured by an approach that gives the location of the shock front varying with the impact time. A comparison of the cell-based finite element model with continuum-based shock models indicates that the shock model based on a material accounting for the post-locking behaviour is much accurate in predicting the shock wave velocity. Finally, stress-strain states ahead of and behind the shock front are obtained. These results provide an explanation in terms of deformation mechanism for the stress reduction at the support end with increasing impact velocity, which was previously observed in experimental and numerical studies.
In the local strain field calculation method, the cut-off radius determines the neighboring nodes of a representative node for calculating the optimal local deformation gradient. Therefore, the local nature and accuracy of the method are dependent on the cut-off radius. The sensitivity of the cut-off radius is investigated to further improve the local strain field calculation method. Two different schemes are first analyzed to determine the suitable cut-off radii by characterizing the heterogeneous deformation of Voronoi honeycombs under uniaxial compression. Then, an optimal scheme is further suggested. It is demonstrated that the local strain field calculation method with the optimal scheme characterizes the heterogeneous deformation of cellular materials reasonable well whether the compression rate is low or high. Finally, the local strain field calculation method is applied to reveal the evolution of the heterogeneous deformation of two different configurations of double-layer cellular claddings (DLCCs) under a linear decaying blast load.2D fields and ID distributions of local engineering strain are calculated. These results interpret the shock wave propagation mechanisms in both claddings and provide a useful understanding in the design of a double-layer cellular cladding.
This thesis further presents a method for the design of a double-layer cellular cladding serving as a protective function against blast load. Two configurations of DLCCs are considered, i.e. cladding with cellular cores of identical density (Cladding-1) and cladding with a higher density cellular core layer close to the blast load (Cladding-2). Shock wave propagation in the two DLCCs is investigated by using the classical one-dimensional shock model. Single shock front propagates in Cladding-1, while double shock fronts propagate in Cladding-2. A closed-form solution of the critical thickness, which is the minimum thickness required to fully absorb the blast load, is given for Cladding-1. Response features of Cladding-2are analyzed and then the critical thickness of Cladding-2is determined by optimizing the layer thicknesses through introducing two constraints. With an equal total mass of cover plates, Cladding-1and Cladding-2are compared with a single-layer cellular cladding (SLCC) in terms of the critical thickness. It is demonstrated that the SLCC is always more efficient than Cladding-1as a protective structure for blast alleviation, while Cladding-2can be more efficient than the SLCC. Moreover, a maximum reduction of the critical thickness of Cladding-2can be achieved by choosing an optimal combination of density ratio of the two cellular cores and mass ratio of the two cover plates. A design method for Cladding-2against blast load is further presented. Finally, cell-based finite element simulations are carried out to verify the analytical predictions about the design of Cladding-2. Comparisons between the cell-based finite element results and the analytical predictions are generally good, and some difference is analyzed through the strain distributions measured by the local strain field calculation method.
引文
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