岩土应变硬化指数理论及其分数阶微积分理论基础_殷德顺

第32卷 第5期 岩 土 工 程 学 报 Vol.32 No.5 2010年 5月 Chinese Journal of Geotechnical Engineering May 2010

岩土应变硬化指数理论及其分数阶微积分理论基础

殷德顺，和成亮，陈 文

(河海大学工程力学系，江苏 南京 210098)

摘 要：应变硬化型岩土的三轴试验应力应变曲线能够表现出不同的弯曲程度，而应力应变曲线的弯曲程度是岩土应变硬化能力的体现，但已有的研究中还没有相应的参数来描述岩土的硬化能力。为了获得反映岩土应变硬化能力的参数，从而有助于了解岩土的塑性性能和指导土体的合理承载，根据Hollomon 提出的描绘金属塑性拉伸变形的指数方程（经验公式），提出了岩土应变硬化指数理论。通过许多三轴试验，发现岩土应变硬化指数理论提出的岩土应力应变关系符合乘幂函数关系的假设能够被验证，岩土的应变硬化指数能够反映岩土的硬化能力。岩土的力学性质介于理想固体和理想流体之间，其应力应变关系既不遵守胡克定律，也不遵守牛顿黏性定律，而是遵守介于它们之间的某种关系。利用分数阶微积分理论给出了恒应变率加载情况下的土应力应变关系。关系式显示应力应变之间也呈乘幂函数关系，说明岩土分数阶应力应变关系能够为岩土应变硬化指数理论提供理论基础。 关键词：应变硬化指数；分数阶微积分；三轴试验；理论基础

中图分类号：TU43 文献标识码：A 文章编号：1000–4548(2010)05–0762–05

作者简介：殷德顺(1972– )，男，副教授，主要从事土的本构模型及基本理论研究。E-mail: yindeshun@ 。

Theory of geotechnical strain hardening index and its rationale from

fractional order calculus

YIN De-shun, HE Cheng-liang, CHEN Wen

(Department of Engineering Mechanics, Hohai University, Nanjing 210098, China)

Abstract : The curvature of stress-strain curves from triaxial tests on harden soil is a reflection of hardening ability, but there isn’t a parameter of hardening ability in geotechnical mechanics. In order to gain such a parameter that can help us to know plasticity and proper bearing capacity of soil, a theory of geotechnical strain hardening index (TAGSHI) in response to exponential equation (an empirical equation) which Hollomon established from experience in metal tensile deformation is developed. Based on a lot of triaxial tests, it is shown that the assumption in TAGSHI is right, which thinks that stress-strain relationship of soil is the power function in triaxial tests, and the strain hardening index may reflect geotechnical hardening ability. As we all know that geotechnical mechanical property should be intermediate between that of an ideal solid and an ideal fluid, so its stress-strain relation should neither follow the Hook’s law nor obey the Newton's law of viscosity, and it should be consistent with the fractional expression ()d ()d t t t ββσε=, (01β

<<). The geotechnical stress-strain relation is derived by

applying the theory of fractional order calculus operator under the condition of loading with constant strain rate. The analytic results show that the geotechnical stress-strain curves exhibit power relation, and it is consistent with the assumption in TAGSHI. This indicates that the fractional expression ()d ()t t t ββσε=, (01β

<<) can give a rationale for TAGSHI.

Key words : strain hardening index; fractional order calculus; triaxial test; rationale

0 前 言

常规三轴试验是研究岩土材料力学性质的重要手段，无论是砂土[1]、黏土[2-3]，还是岩石材料[4]，它们的三轴试验已经积累了大量的试验数据和经验。根据轴向应变达到15%前13()σσ−是否出现峰值，将岩土的应力应变关系分为应变硬化型和应变软化型两类[5]

。而现有的岩土力学只是根据三轴试验结果简单地将岩土材料分为应变硬化型和应变软化型，至今还没

反映岩土应变硬化能力的参数，这使人们不能很好地认识岩土材料的力学性能。比如，常规三轴试验可以得到两种土样的应力应变曲线，如图1（a ）所示（一种土样为实线，另一种土样为虚线），由于它们破坏时应力相同，所以它们可以得到相同的莫尔应力圆，也

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基金项目：中央高校基本科研业务专项资金资助项目 收稿日期：2009–01–16