基于广义位势理论的土的本构模型及其初步应用
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摘要
土的本构模型是现代土力学的基本方程,土的本构模型的研究对现代土力学的发展具有重要意义。现有用于建立土的本构模型的理论主要是基于传统的金属材料本构理论,然而土的力学特性与金属材料有很大的不同,直接应用是不适合的,即使在其基础上进行修正,但由于传统理论基本假设及理论基础的限制,也还是存在一定的局限性。
广义位势理论从数学角度出发建立了土的统一的本构模型理论体系,可克服金属材料本构理论的不足而又同时包含了传统理论作为其特例,从而为土的本构模型研究提供新的和更一般的理论。同时,由于土的力学性质的复杂性,还有不少问题有待于进一步深入地研究。本文基于广义位势理论,针对目前土的本构模型中存在的一些主要问题展开研究,提出了几个简化的实用本构模型并进行了初步的应用。主要研究内容及结论如下：
(1)从广义位势理论出发,揭示了传统位势理论的数学实质是在一定的假设条件下的坐标变换问题,因而传统位势理论可看作为广义位势理论在一定假设条件下的特例；并进一步分析了传统塑性位势理论在描述土的塑性应变增量方向非唯一性以及土的非共轴性等问题上的局限性,指出广义位势理论可为解决这些问题提供更为广阔的理论基础。
(2)针对剑桥模型中采用的“能量方程”假设所带来的模糊性,从数学角度出发重新构建并发展剑桥模型,建立了基于广义位势理论的类剑桥模型。该模型既保持了原有剑桥模型参数确定简便的优点,同时又具有了更明确的数学基础；此外,该模型具有更大的灵活性,通过引入一个新的参数,可建立考虑剪胀性的类剑桥模型。试验验证结果表明：类剑桥模型计算结果与试验结果吻合较好,且能够反映剪胀性,效果优于传统的剑桥模型和修正剑桥模型。
(3)针对传统塑性位势理论在描述土的塑性应变增量方向非唯一性以及土的非共轴性问题上的局限性,根据基于广义位势理论的拟弹性弹塑性模型提出了解决方法。该模型将塑性应变增量按其主要构成特性分解为方向与应力增量方向相关的拟弹性部分和方向由总应力状态确定的符合正交流动法则的纯塑性部分,这样分解后建立的模型更符合土的实际变形机理,并可应用于解决土的塑性应变增量方向非唯一性以及土的非共轴性问题。试验验证结果表明：基于广义位势理论的拟弹性弹塑性模型能够同时反映土的塑性应变增量方向的唯一性(高应力水平)和非唯一性(低应力水平),以及合理地描述主应力轴旋转过程中土的非共轴性,结果更符合实际。
(4)针对邓肯—张模型基于广义胡克定律的理论限制以及采用双曲线函数拟合试验曲线的局限性,根据基于广义位势理论的数值弹塑性模型提出了解决方法。该模型既保留了邓肯—张模型在参数确定方面的简单性,又可弥补邓肯—张模型在反映土体剪胀性方面的缺陷。试验验证结果表明：基于广义位势理论的数值弹塑性模型计算结果与试验结果吻合良好,且能够反映剪胀性,从而初步证明了该模型的合理性。
(5)基于FLAC3D软件实现了基于广义位势理论的类剑桥模型、拟弹性弹塑性模型和数值弹塑性模型的二次开发,并对三峡二期围堰等几个工程问题进行了分析。数值模拟结果表明：与共轴模型相比,基于广义位势理论的拟弹性弹塑性模型能够考虑主轴旋转对土体变形的影响；与邓肯—张模型相比,基于广义位势理论的数值弹塑性模型能够反映土体的剪胀特性,其计算结果与实测结果具有更好的一致性,从而进一步验证了该模型的合理性。
The constitutive model of soils is the basic equation of the modern soil mechanics, and the study on it has important significance for the development of the modern soil mechanics. At present, most of the theories which used to establish the constitutive model of soils are based on the traditional theories for metal materials. However, the mechanical properties of soils and metal materials are very different, which makes the direct application of the traditional theories on the constitutive model of soils exist great limitation. Even though making some modifications based on the traditional theories, the limitations also exist due to some basic assumptions in the traditional theories.
The generalized potential theory, in wihch the constitutive model of soils is created directly by mathematical methods, and the deficiencies in the traditional theories for metal materials can be overcomed and the traditional theories can be regarded as its special cases. It gives a new theory to the study of constitutive model of soils. Meanwhile, due to the complexity of the mechanical properties of soils, many problems need further study. Therefore, according to the major problems in the present study of constitutive model of soils, some research works are carried out based on the generalized potential theory in this paper, and several constitutive models of soil are established and applied. The main contents and conclusions are summarized as follows:
(1) The ansysis of mathematical essential of the traditional potential theories based on the generalized potential theory shows that the traditional potential theories are actually coordinate transformation under certain assumptions, and they can be regarded as special cases of the generalized potential theory. What is more, further analysis shows that the traditional potential theories can not describe the non-uniqueness of the direction of plastic strain increment and the non-coaxiality of soils, while the generalized potential theory can give a new theory to the study on these problems.
(2) Aiming at the fuzziness brought by the energy equation hypothesis in the Cam-clay model, a similar Cam-clay model was established by mathematical methods base on the generalized potential theory. Parameters of the similar Cam-clay model can be determined as conveniently as the Cam-clay model, and the mathematical principle of similar Cam-clay model is more clear. Meanwhile, by introducing a new parameter, the similar Cam-clay model can describe the dilatancy of soil. The results of test verification show that the calculated results of the similar Cam-clay model agree well with test ones and the dilatancy of sand can also be described, which are better than the calculated results of Cam-clay model and the modified Cam-clay model.
(3) Aiming at the limitations of the traditional potential theories on describing the non-uniqueness of the direction of plastic strain increment and the non-coaxiality of soils, solutions were proposed acrroding to the quasi-elastic-plastic constitutive model based on the generalized potential theory, in which the traditional plastic strain increment was decomposed into quasi-elastic part and pure-plastic part. The quasi-elastic part obeys elastic rule and the pure-plastic part obeys the traditional plasticity theory, which is reasonable and convenient, and it can also reflect the non-uniqueness of the direction of plastic strain increment and the non-coaxiality of soils. The results of test verification show that the quasi-elastic-plastic constitutive model based on the generalized potential theory can reflect the uniqueness (in high stress level) and the non-uniqueness (in low stress level) of the direction of plastic strain increment of soil simultaneously, and the non-coaxiality of soil during the rotation of principal stress axis can also be reflected, which agree with the actual.
(4) Aiming at the limitations of Duncan-Chang model which is based on the generalized Hooke's law and adopts hyperbola hypothesis, solutions were proposed acrroding to the numerical elastic-plastic constitutive model based on the generalized potential theory, in which parameters can be determined as conveniently as Duncan-Chang model and the dilatancy of soils can also be described. The results of test verification show that the calculated results of the numerieal elastic-plastic constitutive model based on the generalized potential theory agree well with test ones and the dilatancy of soils can also be described, which proves the rationality of the numerieal elastic-plastic constitutive model.
(5) Based on FLAC3D, the secondary developments of the similar Cam-clay model and the quasi-elastic-plastic constitutive model and the numerieal elastic-plastic constitutive model based on the generalized potential theory were implemented, and then they were applied to the ansysis of the second stage cofferdam of three gorges project etc. The results of numerical simulation show that, compared to the coaxial model, the quasi-elastic-plastic constitutive model based on the generalized potential theory can reflect the influence of the rotation of principal stresses axis to the deformation of soil. While compared to Duncan-Chang model, the numerieal elastic-plastic constitutive model based on the generalized potential theory can describe the dilatancy of soil and the calculated results of it are more agree with the measured results, which furtherly proves the rationality of the proposed model.
广义位势理论从数学角度出发建立了土的统一的本构模型理论体系,可克服金属材料本构理论的不足而又同时包含了传统理论作为其特例,从而为土的本构模型研究提供新的和更一般的理论。同时,由于土的力学性质的复杂性,还有不少问题有待于进一步深入地研究。本文基于广义位势理论,针对目前土的本构模型中存在的一些主要问题展开研究,提出了几个简化的实用本构模型并进行了初步的应用。主要研究内容及结论如下：
(1)从广义位势理论出发,揭示了传统位势理论的数学实质是在一定的假设条件下的坐标变换问题,因而传统位势理论可看作为广义位势理论在一定假设条件下的特例；并进一步分析了传统塑性位势理论在描述土的塑性应变增量方向非唯一性以及土的非共轴性等问题上的局限性,指出广义位势理论可为解决这些问题提供更为广阔的理论基础。
(2)针对剑桥模型中采用的“能量方程”假设所带来的模糊性,从数学角度出发重新构建并发展剑桥模型,建立了基于广义位势理论的类剑桥模型。该模型既保持了原有剑桥模型参数确定简便的优点,同时又具有了更明确的数学基础；此外,该模型具有更大的灵活性,通过引入一个新的参数,可建立考虑剪胀性的类剑桥模型。试验验证结果表明：类剑桥模型计算结果与试验结果吻合较好,且能够反映剪胀性,效果优于传统的剑桥模型和修正剑桥模型。
(3)针对传统塑性位势理论在描述土的塑性应变增量方向非唯一性以及土的非共轴性问题上的局限性,根据基于广义位势理论的拟弹性弹塑性模型提出了解决方法。该模型将塑性应变增量按其主要构成特性分解为方向与应力增量方向相关的拟弹性部分和方向由总应力状态确定的符合正交流动法则的纯塑性部分,这样分解后建立的模型更符合土的实际变形机理,并可应用于解决土的塑性应变增量方向非唯一性以及土的非共轴性问题。试验验证结果表明：基于广义位势理论的拟弹性弹塑性模型能够同时反映土的塑性应变增量方向的唯一性(高应力水平)和非唯一性(低应力水平),以及合理地描述主应力轴旋转过程中土的非共轴性,结果更符合实际。
(4)针对邓肯—张模型基于广义胡克定律的理论限制以及采用双曲线函数拟合试验曲线的局限性,根据基于广义位势理论的数值弹塑性模型提出了解决方法。该模型既保留了邓肯—张模型在参数确定方面的简单性,又可弥补邓肯—张模型在反映土体剪胀性方面的缺陷。试验验证结果表明：基于广义位势理论的数值弹塑性模型计算结果与试验结果吻合良好,且能够反映剪胀性,从而初步证明了该模型的合理性。
(5)基于FLAC3D软件实现了基于广义位势理论的类剑桥模型、拟弹性弹塑性模型和数值弹塑性模型的二次开发,并对三峡二期围堰等几个工程问题进行了分析。数值模拟结果表明：与共轴模型相比,基于广义位势理论的拟弹性弹塑性模型能够考虑主轴旋转对土体变形的影响；与邓肯—张模型相比,基于广义位势理论的数值弹塑性模型能够反映土体的剪胀特性,其计算结果与实测结果具有更好的一致性,从而进一步验证了该模型的合理性。
The constitutive model of soils is the basic equation of the modern soil mechanics, and the study on it has important significance for the development of the modern soil mechanics. At present, most of the theories which used to establish the constitutive model of soils are based on the traditional theories for metal materials. However, the mechanical properties of soils and metal materials are very different, which makes the direct application of the traditional theories on the constitutive model of soils exist great limitation. Even though making some modifications based on the traditional theories, the limitations also exist due to some basic assumptions in the traditional theories.
The generalized potential theory, in wihch the constitutive model of soils is created directly by mathematical methods, and the deficiencies in the traditional theories for metal materials can be overcomed and the traditional theories can be regarded as its special cases. It gives a new theory to the study of constitutive model of soils. Meanwhile, due to the complexity of the mechanical properties of soils, many problems need further study. Therefore, according to the major problems in the present study of constitutive model of soils, some research works are carried out based on the generalized potential theory in this paper, and several constitutive models of soil are established and applied. The main contents and conclusions are summarized as follows:
(1) The ansysis of mathematical essential of the traditional potential theories based on the generalized potential theory shows that the traditional potential theories are actually coordinate transformation under certain assumptions, and they can be regarded as special cases of the generalized potential theory. What is more, further analysis shows that the traditional potential theories can not describe the non-uniqueness of the direction of plastic strain increment and the non-coaxiality of soils, while the generalized potential theory can give a new theory to the study on these problems.
(2) Aiming at the fuzziness brought by the energy equation hypothesis in the Cam-clay model, a similar Cam-clay model was established by mathematical methods base on the generalized potential theory. Parameters of the similar Cam-clay model can be determined as conveniently as the Cam-clay model, and the mathematical principle of similar Cam-clay model is more clear. Meanwhile, by introducing a new parameter, the similar Cam-clay model can describe the dilatancy of soil. The results of test verification show that the calculated results of the similar Cam-clay model agree well with test ones and the dilatancy of sand can also be described, which are better than the calculated results of Cam-clay model and the modified Cam-clay model.
(3) Aiming at the limitations of the traditional potential theories on describing the non-uniqueness of the direction of plastic strain increment and the non-coaxiality of soils, solutions were proposed acrroding to the quasi-elastic-plastic constitutive model based on the generalized potential theory, in which the traditional plastic strain increment was decomposed into quasi-elastic part and pure-plastic part. The quasi-elastic part obeys elastic rule and the pure-plastic part obeys the traditional plasticity theory, which is reasonable and convenient, and it can also reflect the non-uniqueness of the direction of plastic strain increment and the non-coaxiality of soils. The results of test verification show that the quasi-elastic-plastic constitutive model based on the generalized potential theory can reflect the uniqueness (in high stress level) and the non-uniqueness (in low stress level) of the direction of plastic strain increment of soil simultaneously, and the non-coaxiality of soil during the rotation of principal stress axis can also be reflected, which agree with the actual.
(4) Aiming at the limitations of Duncan-Chang model which is based on the generalized Hooke's law and adopts hyperbola hypothesis, solutions were proposed acrroding to the numerical elastic-plastic constitutive model based on the generalized potential theory, in which parameters can be determined as conveniently as Duncan-Chang model and the dilatancy of soils can also be described. The results of test verification show that the calculated results of the numerieal elastic-plastic constitutive model based on the generalized potential theory agree well with test ones and the dilatancy of soils can also be described, which proves the rationality of the numerieal elastic-plastic constitutive model.
(5) Based on FLAC3D, the secondary developments of the similar Cam-clay model and the quasi-elastic-plastic constitutive model and the numerieal elastic-plastic constitutive model based on the generalized potential theory were implemented, and then they were applied to the ansysis of the second stage cofferdam of three gorges project etc. The results of numerical simulation show that, compared to the coaxial model, the quasi-elastic-plastic constitutive model based on the generalized potential theory can reflect the influence of the rotation of principal stresses axis to the deformation of soil. While compared to Duncan-Chang model, the numerieal elastic-plastic constitutive model based on the generalized potential theory can describe the dilatancy of soil and the calculated results of it are more agree with the measured results, which furtherly proves the rationality of the proposed model.
引文
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