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摘 要：傳统不同模量理论中基于主应力方向建立的本构方程，仅能表述主应力方向的应力应变关系，并未体现出其他方向的应力应变特性，不能有效表征拉压不同模量问题的力学本质.基于此，在主应力方向的本构方程基础上，利用应力及应变的转轴公式，推导了基于不同直角坐标系下的拉压不同模量本构方程的具体形式，也即广义弹性定律.经理论验证，此广义弹性定律揭示了拉压不同模量问题既是非线性问题也体现出各向异性的力学性质;并且在拉压模量相等时可以回退到经典弹性理论本构方程，而基于主应力方向建立的本构方程是广义弹性定律中的特例.针对不同模量理论中不甚明晰的剪切模量和泊松比-弹性模量比值的假设，应用所得到的广义弹性定律对纯剪应力状态进行了力学分析，分析表明：在基于最大或最小剪应力方向的直角坐标系下，剪应力与剪应变成线性关系，剪切模量保持不变;并结合微元体纯剪变形的几何关系，证明了假设即拉泊松比与拉模量之比等于压泊松比与压模量之比在纯剪受力状态下是自然满足的.
关键词：弹性理论;不同模量;本构方程;主应力;纯剪
中图分类号：O343 文献标志码：A
文章编号：1674—2974（2019）01—0093—08
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.