The Intrinsic Dependence Of Poisson’s Ratio On Stretch And Rotation

Poisson’s ratio, together with Young’s modulus, are two independent moduli of elasticity to characterize the elastic behaviour of isotropic materials. Poisson’s ratio d-iffers from one material to another depending on the microstructure. So the research of the mechanism of Poisson’s ratio focuses on the deformation of microstructure, espe-cially the expansion and rotation. Beam model and mass-spring model are introduced to simulate the deformation of structures when stretched. The results of this two mod-els are identical while calculating expansion but different while bending because of the different resistance methods. In contrast to the different bending stiffness, the corre-sponding dimension is the same. No matter which model is chosen, the conclusion will be consistent.Starting from the basic structures, orientation and material properties as well as coordination number are obtained as the factors of Poisson’s ratio though simulation of re-entrant unit structure. The formulation of Poisson’s ratio is used to further explore the mechanism and a new calculation by adapting the Lagrange strain of radius of gyra-tion is put forward. The new calculation is the same with the traditional way while the structure is big enough. Comparison between the new calculation and another one using the Lagrange strain of border is given, which show the advantage of the new calculation. The influence of expansion and rotation on Poisson’s ratio is acquired deducing from the new calculation. Expansion and rotation of connections between any two points are needed according to the influence which indicates a modified way. A negative Pois-son’s ratio is gained when the modified expansion is lager than the modified rotation, and a positive Poisson’s ratio occurs when reversed. Material properties have direct effects on beams which correspond edges actually connected between two nodes. As a conclusion, material properties control the expansion and rotation of beams, which affect the modified expansion and rotation of connections of any two nodes and finally decide Poisson’s ratio.Eigenmode, a classical method to study deformation, is employed to explore the mechanism of Poisson’s ratio. Eigenmodes of network based on beam model are calcu-lated and the deformation of network with uniaxial tensile is mapped into each eigen-mode. Deformation ratio of eigenmode is proposed and both calculations of Poisson’s ratio can acquire the function that it is weighted sum of deformation ratio. The weight is a product of contribution of each eigenmode and the strain ratio of each eigenmode to the whole model while the strain must be consistent with the calculation of Poisson’s ratio. The eigenmode can be treated as a basic deformation while the conclusion of Poisson’s ratio can also be used here, which means that a negative deformation ratio is gained when modified expansion is lager than modified rotation and vice versa.Disordered triangle network, the most famous complex structure, together with Voronoi network which has great great application in many research areas, are employed to verify the relevant conclusions. The results of this two structures under beam model and mass-spring model reach consensus and different calculations of Poisson’s ratio are exactly the same. The distribution of modified expansion and rotation, which is divided into two parts of positive and negative Poisson’s ratio by the line representing expansion equals to rotation, is showed under different material properties. Material properties decide the trend of expansion and rotation of beams and the connections have the same trend. The contribution of eigenmodes is showed to verify the relationship between Poisson’s ratio and deformation ratio and the distribution of modified expansion and rotation of each eigenmode is given at the same time.