基于Cosserat连续体模型的颗粒材料宏细观力学行为数值模拟
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摘要
颗粒材料与人们的日常生活息息相关,广泛存在于自然界并在实际工程中被大量地应用,例如粒状药剂、砂砾、堆石料等。颗粒材料是由大量离散固体颗粒构成的,具有非常复杂的性质,其力学行为的理论研究和数值模拟受到众多学者的广泛关注。
剪胀性是颗粒材料的重要宏观力学行为之一,一般通过引入剪胀角来表征其影响。在进行工程计算分析时,对剪胀角ψ一般有三种处理方式：(1)ψ=0°;(2)ψ=?μ,?μ为材料的内摩擦角；(3)0°<ψ≡const<φμ。以上三种处理方式都有着各自的弊端：第一种处理方式没有考虑颗粒材料的剪胀性；第二种处理方式则夸大了颗粒材料的剪胀性,并且与塑性能量耗散理论也存在一定的矛盾；第三种处理方式是前两种处理方式的折中,是一种过于依赖工程经验的方法。同时以上三种处理方式中剪胀角均为常数,这将导致剪胀会随剪切应变的增大而呈线性增加,而此与颗粒材料达到临界状态后其塑性体积不再增加的实际情况也是不相符的。本文借鉴了Houlsby提出的剪胀角相关公式,将其与Drucker-Prager准则结合并引入到了Cosserat连续体模型之中,形成了一个能考虑剪胀角演化的宏观连续体模型并运用Fortran语言独立开发了程序代码将其数值实现,最后通过数值算例对颗粒材料结构的承载力以及应变局部化现象进行了研究。
同时颗粒破碎也是颗粒材料的一个重要特性,其对颗粒材料的宏观力学响应也有着影响。颗粒破碎所引起的最直观变化为颗粒粒径的改变,经典连续体模型大多无法对其加以描述,而Cosserat连续体模型中包含的特征长度参数则在一定程度上反映了细观结构内的平均颗粒粒径。本文中借鉴Hardin提出的颗粒破碎相关公式,将表征颗粒材料破碎程度的相对破碎率与Cosserat连续体模型中的特征长度参数关联了起来,同时还提出了破碎应力阈值对相对破碎率的计算进行了适当修正,形成了一个能考虑颗粒破碎的宏观连续体模型并运用Fortran语言独立开发了程序代码将其数值实现,最后通过数值算例对颗粒材料结构的承载力以及应变局部化现象进行了研究。
此外,颗粒材料多尺度模型的建立为颗粒材料的研究提供了一种新的途径。本文中使用了宏观Cosserat连续体模型-细观离散颗粒模型两尺度模型,该模型在宏观尺度上依赖于宏观有限元网格,能有效地解决规模较大的问题；同时在细观尺度上将颗粒材料视为离散颗粒集合体,以便于更真实地描述颗粒材料的离散特性。此种方法是由计算均匀化理论发展而来,其核心为基于表征元的宏细观信息的传递,而细观数值样本尺寸的选取合适与否则关系到细观数值样本是否能被视作为表征元。本文基于Miehe提出的宏细观信息的传递格式详细研究了细观数值样本尺寸对颗粒材料结构的宏观刚度、极限承载力以及残余承载力产生的影响,并根据数值分析结果提出了相应的指数-对数型拟合公式以便于确定合适的细观数值样本尺寸,同时还研究了加载过程中细观尺度上细观数值样本构形及位移残差场的演化。
Granular materials are closely linked to human beings' daily life and are widely used in engineering, such as gravel, grainy pharmacy and so on. Generally speaking, granular materials are composed of randomly packed discrete particles partially or fully filled with void fluids and they have many complex charateristics. So many researchers devote themselves to theoretical and numerical studies of mechanical behaviors of granular materials.
Many features of granular materials, such as the reduction of friction angle with increasing stress level, are related to dilatancy; therefore the understanding of dilatancy is the key to investigating behaviors of granular materials. In most engineering analysis, effect of dilatancy is usually taken into account by introducing the dilatancy angle. Dilatancy angle is generally considered by the following approaches:(1) assuming that the dilatancy angle is equal to zero;(2) assuming that the dilatancy angle is equal to the internal friction angle;(3) assuming that the dilatancy angle is a constant in the range from zero to the internal fricition angle. However, these approaches all have obvious drawbacks:the first method completely ignores dilatancy of granular materials; the second method expands dilatancy of granular materials and it also conflicts with the plastic energy dissipation theory; the last method is a way of over-reliance on engineering experience. While dilatancy angle in all three methods is kept as a constant, which causes that dilatancy increases linearly with the shear strain increasing. And this phenomenon is not consistent with that the plastic volume does not increase after achieving the critical state. In this paper, the specify formula about dilatancy angle given by Houlsby is introduced into plastic potential function in conjunct with Drucker-Prager yield function for Cosserat continuum model for granular materials. Based on two simple numerical examples, it has been show that evolution of dilatancy angle has some effect on the bearing capacity and strain localization. For the bearing capacity, with evolution of dilatancy angle, the larger dilatancy angle will lead to larger bearing capacity, and after the peak of the bearing capacity it will exhibit obvious nonlinear behaviors. For the strain localization, especially for slope stability problem, the larger dilatancy angle causes a more obvious slide band.
At the same time, particles' breakage has an important effect on macroscopic mechanical behaviors of granular materials. Cosserat continuum model contains the internal length scale which represents the average particles' size of micro structure in granular materials to some extent. So the internal length scale can reflect changes of particles' radius caused by particles' breakage. In this paper, an elastoplastic model combined with an experiential crushing equation is suggested for crushable granular materials based on Cosserat continuum model, Hardin's definition of relative breakage Br, which is used to quantify the extent of crushing, can be obtained from the crushing equation according to normal crushing stress. Numerical examples mainly focus on the effect of particle crushing on the bearing capacity and localization of plastic strain. Numerical results illustrate that particles crush mainly in shear band, and shear band obviously becomes narrow and the equivalent plastic strain gradient increases when considering crushing.
In addition, multiscale method for granular materials provides a new approach for the study of granular materials. The macro-micro mechanical behavior of granular materials is investigated based on a two-scale method (Finite Element Method-Discrete Element Method, FEM-DEM). In this method, the macro-stresses are obtained from the average of contact forces between discrete particles in RVE (Representative Volume Element) and the macro deformation provides the boundary condition for RVE, meanwhile the micro behavior of granular materials is modeled by the DEM. So RVE is the key point for this method and it needs to choose a micro structure which has an appropriate size to become RVE. Part of this paper's content is mainly focused on the influence of the size of micro structure on the deformation stiffness, the bearing capacity and the residual strength of the numerical model and suggests the corresponding fitting formulas according to the results. In addition, the micro structure's configuration and the particles' displacement fluctuation in the loading process are also concerned. Comparing the results under the same vertical loading displacement, it can be found that the macro deformation is closely associated with the evolution of the micro structure:because the micro structure's configuration and closely packed form both don't change obviously before the ultimate bearing capacity, the bearing capacity increases as the load displacement's increasement. After the ultimate bearing capacity, the micro structure's area becomes bigger apparently and closely packed form gradually disappears. As results of these changes, the bearing capacity has a significant softening and the particles' displacement fluctuation includes some slip bands.
剪胀性是颗粒材料的重要宏观力学行为之一,一般通过引入剪胀角来表征其影响。在进行工程计算分析时,对剪胀角ψ一般有三种处理方式：(1)ψ=0°;(2)ψ=?μ,?μ为材料的内摩擦角；(3)0°<ψ≡const<φμ。以上三种处理方式都有着各自的弊端：第一种处理方式没有考虑颗粒材料的剪胀性；第二种处理方式则夸大了颗粒材料的剪胀性,并且与塑性能量耗散理论也存在一定的矛盾；第三种处理方式是前两种处理方式的折中,是一种过于依赖工程经验的方法。同时以上三种处理方式中剪胀角均为常数,这将导致剪胀会随剪切应变的增大而呈线性增加,而此与颗粒材料达到临界状态后其塑性体积不再增加的实际情况也是不相符的。本文借鉴了Houlsby提出的剪胀角相关公式,将其与Drucker-Prager准则结合并引入到了Cosserat连续体模型之中,形成了一个能考虑剪胀角演化的宏观连续体模型并运用Fortran语言独立开发了程序代码将其数值实现,最后通过数值算例对颗粒材料结构的承载力以及应变局部化现象进行了研究。
同时颗粒破碎也是颗粒材料的一个重要特性,其对颗粒材料的宏观力学响应也有着影响。颗粒破碎所引起的最直观变化为颗粒粒径的改变,经典连续体模型大多无法对其加以描述,而Cosserat连续体模型中包含的特征长度参数则在一定程度上反映了细观结构内的平均颗粒粒径。本文中借鉴Hardin提出的颗粒破碎相关公式,将表征颗粒材料破碎程度的相对破碎率与Cosserat连续体模型中的特征长度参数关联了起来,同时还提出了破碎应力阈值对相对破碎率的计算进行了适当修正,形成了一个能考虑颗粒破碎的宏观连续体模型并运用Fortran语言独立开发了程序代码将其数值实现,最后通过数值算例对颗粒材料结构的承载力以及应变局部化现象进行了研究。
此外,颗粒材料多尺度模型的建立为颗粒材料的研究提供了一种新的途径。本文中使用了宏观Cosserat连续体模型-细观离散颗粒模型两尺度模型,该模型在宏观尺度上依赖于宏观有限元网格,能有效地解决规模较大的问题；同时在细观尺度上将颗粒材料视为离散颗粒集合体,以便于更真实地描述颗粒材料的离散特性。此种方法是由计算均匀化理论发展而来,其核心为基于表征元的宏细观信息的传递,而细观数值样本尺寸的选取合适与否则关系到细观数值样本是否能被视作为表征元。本文基于Miehe提出的宏细观信息的传递格式详细研究了细观数值样本尺寸对颗粒材料结构的宏观刚度、极限承载力以及残余承载力产生的影响,并根据数值分析结果提出了相应的指数-对数型拟合公式以便于确定合适的细观数值样本尺寸,同时还研究了加载过程中细观尺度上细观数值样本构形及位移残差场的演化。
Granular materials are closely linked to human beings' daily life and are widely used in engineering, such as gravel, grainy pharmacy and so on. Generally speaking, granular materials are composed of randomly packed discrete particles partially or fully filled with void fluids and they have many complex charateristics. So many researchers devote themselves to theoretical and numerical studies of mechanical behaviors of granular materials.
Many features of granular materials, such as the reduction of friction angle with increasing stress level, are related to dilatancy; therefore the understanding of dilatancy is the key to investigating behaviors of granular materials. In most engineering analysis, effect of dilatancy is usually taken into account by introducing the dilatancy angle. Dilatancy angle is generally considered by the following approaches:(1) assuming that the dilatancy angle is equal to zero;(2) assuming that the dilatancy angle is equal to the internal friction angle;(3) assuming that the dilatancy angle is a constant in the range from zero to the internal fricition angle. However, these approaches all have obvious drawbacks:the first method completely ignores dilatancy of granular materials; the second method expands dilatancy of granular materials and it also conflicts with the plastic energy dissipation theory; the last method is a way of over-reliance on engineering experience. While dilatancy angle in all three methods is kept as a constant, which causes that dilatancy increases linearly with the shear strain increasing. And this phenomenon is not consistent with that the plastic volume does not increase after achieving the critical state. In this paper, the specify formula about dilatancy angle given by Houlsby is introduced into plastic potential function in conjunct with Drucker-Prager yield function for Cosserat continuum model for granular materials. Based on two simple numerical examples, it has been show that evolution of dilatancy angle has some effect on the bearing capacity and strain localization. For the bearing capacity, with evolution of dilatancy angle, the larger dilatancy angle will lead to larger bearing capacity, and after the peak of the bearing capacity it will exhibit obvious nonlinear behaviors. For the strain localization, especially for slope stability problem, the larger dilatancy angle causes a more obvious slide band.
At the same time, particles' breakage has an important effect on macroscopic mechanical behaviors of granular materials. Cosserat continuum model contains the internal length scale which represents the average particles' size of micro structure in granular materials to some extent. So the internal length scale can reflect changes of particles' radius caused by particles' breakage. In this paper, an elastoplastic model combined with an experiential crushing equation is suggested for crushable granular materials based on Cosserat continuum model, Hardin's definition of relative breakage Br, which is used to quantify the extent of crushing, can be obtained from the crushing equation according to normal crushing stress. Numerical examples mainly focus on the effect of particle crushing on the bearing capacity and localization of plastic strain. Numerical results illustrate that particles crush mainly in shear band, and shear band obviously becomes narrow and the equivalent plastic strain gradient increases when considering crushing.
In addition, multiscale method for granular materials provides a new approach for the study of granular materials. The macro-micro mechanical behavior of granular materials is investigated based on a two-scale method (Finite Element Method-Discrete Element Method, FEM-DEM). In this method, the macro-stresses are obtained from the average of contact forces between discrete particles in RVE (Representative Volume Element) and the macro deformation provides the boundary condition for RVE, meanwhile the micro behavior of granular materials is modeled by the DEM. So RVE is the key point for this method and it needs to choose a micro structure which has an appropriate size to become RVE. Part of this paper's content is mainly focused on the influence of the size of micro structure on the deformation stiffness, the bearing capacity and the residual strength of the numerical model and suggests the corresponding fitting formulas according to the results. In addition, the micro structure's configuration and the particles' displacement fluctuation in the loading process are also concerned. Comparing the results under the same vertical loading displacement, it can be found that the macro deformation is closely associated with the evolution of the micro structure:because the micro structure's configuration and closely packed form both don't change obviously before the ultimate bearing capacity, the bearing capacity increases as the load displacement's increasement. After the ultimate bearing capacity, the micro structure's area becomes bigger apparently and closely packed form gradually disappears. As results of these changes, the bearing capacity has a significant softening and the particles' displacement fluctuation includes some slip bands.
引文
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