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

#### 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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ZHANG H B, ZHANG L,GAO Q. An efficient computational method for mechanical analysis of bimodular structures based on parametric variational

GIANLUCA M. A nonlinear elastic model for isotropic materials with different behavior in tension and compression [J]. Transpiration of the ASME, 1982, 104 (26):26—28

QU C Z. Deformation of geocell with different tensile and compressive modulus [J]. Journal of Geotechnical and Geoenvironmental Engineering, 2009, 14:1—14.

TIMOSHENDO S. Strength of materials, part #: advanced theory and problems [M].2nd ed. Princeton:Van Nostrand,1941:1—100.

AMBARTSUMYAN S A. The basis equations of the theory of elasticity for materials with different tensile and compressive strengths [J]. Mekhanika Tverdogo Tela, 1966 (2):44—53.

JONES R M. Stress –strain relations for materials with different moduli in tension and compression [J]. AIAA Journal, 1977, 15 (1):16—23.

VIJAYADUMAR D, RAO D P. Stress–strain relation for composites with different stiffness in tension and compression–a new model [J]. International Journal of Computational Mechanics, 1987, 1(2): 167—175.

AMBARTSUMYAN C A. The different modulus of elasticity [M]. Translated by BU R F, SHANG Y Z. Beijing: Chinese Railway Press, 1986:1—71 (In Chinese)

YAO B J, YE Z M. Internal force analysis due to the supporting translation statistically indeterminate structures for different elastics modulus [J]. Journal of Shanghai University (English Edition), 2004, 8(3): 274—280.

YAO B J, YE Z M. Analytical solution of bending –compression solemn using different tension–compression modulus [J]. Applied Mathematics and Mechanics, 2004, 25 (9):901—909 (In Chinese)

ZHANG L F,YAO WJ. Biaxial bending of rectangular plates witch different modulus [J]. Journal of Shanghai University(Natural Science Edition), 2017, 23 (1):128—137(. In Chinese)

HE X T,CHEN S L,SUN J Y. Elasticity solution of simple beams with different modulus under uniformly distributed load[J]. Engineering Mechanics,2007,24(10):51—56 (In Chinese)

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