# Theoretical Study on Generalized Elastic Laws of Elastic Theory with Different Modulus

PAN Qinxue, ZHENG Jianlong, BEN Pihua

#### Abstract

In classical elasticity theory with different modulus, the constitutive equations based on the direction of principal stress can only represent the relationship between the principal stress and principal strain in the main stress direction and cannot reflect the stress-strain behavior in other directions, and the mechanical essence of the problem on different modulus in tension and compression cannot be characterized effectively. Therefore, according to the constitutive equations based on the direction of principal stress，the generalized elastic laws were deduced by the rotation formulas of stress and strain under different Cartesian coordinate system, which are constitutive equations with different modulus in tension and compression. With theoretical verification, both the nonlinearity and anisotropy property of bi-modulus materials were revealed by the generalized elastic laws. Furthermore, it can also degenerate to the classical bi-modulus elasticity law, which implies that the constitutive law for material with different modulus in tension and compression is special cases of the obtained results. With respect to the indistinct issues about the shear modulus and the assumption of the ratios between Poisson's ratio and Young's modulus, bimodulus material point under pure shear state was investigated. It is shown that, in the rectangular coordinate system based on the maximum or minimum shear stress direction, the relation between shear stress and shear strain is linear. In other words, the shear modulus keeps invariant；besides，the hypothesis is proved that the ratio of tensile Poisson's ratio to tensile modulus is equal to the ratio of compressive Poisson′s ratio to compressive modulus under pure shear state, combining with the geometric relationship of pure shear deformation in differential element.

Keywords: elastic theory,  different modulus,  constitutive equations,  principal stress,  pure shear

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