细观偶应力/应变梯度弹塑性理论的几个基本问题
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摘要
传统连续体理论引入均匀化假定,认为材料性质从宏观一直延伸到微细观保持不变。这种理论已被成功用于宏观结构力学性能分析。但是,当尺寸降至微细观尺度,材料微缺陷(如位错、微孔洞、微夹杂)的作用开始显现,这导致了材料性质的不均匀性,在实际上表现为尺寸效应。此时,传统连续体理论不再适用。此外,采用传统理论有限元法计算软化材料的局部化问题会出现病态的网格依赖性现象。要想解决尺寸效应和网格依赖性问题,必须求诸于其他理论。
在Cosserat理论的基础上,众多学者发展出了多种细观偶应力／应变梯度理论。这些理论均引入了材料长度参数,可以衡量材料微缺陷的尺寸和软化材料局部化变形的宽度,因此可以成功地解决尺寸效应和网格依赖性问题。但是,细观偶应力／应变梯度理论存在着几个基本问题：
(1)如何构建微细观尺度的连续体理论没有统一的认识,目前的偶应力／应变梯度理论不像弹性力学那样拥有标准的理论框架,它们多是基于不同思想建立起来的。这导致了细观偶应力／应变梯度理论类型繁多,目前常见的理论有十多种,工程应用难以取舍；
(2)除了个别的基于细观机制的应变梯度理论,多数理论包含的材料长度参数没有明确的物理意义,它们的特点还有待进一步研究。目前常采用实验手段标定材料长度参数,但还缺乏精确稳定的偶应力／应变梯度弹塑性解析解作为标定的理论依据。
本文就这两个基本问题展开研究,详细对比了各种理论的特点以便作为工程使用的参考,采用理论推导和数值计算的手段分析了材料长度参数的特点,并构造了两种微弯曲偶应力弹塑性解析解作为标定材料长度参数的依据。各章的具体安排如下：
第一章详细给出了细观偶应力／应变梯度理论的发展近况,包括理论的工程背景、各理论的特点简介、各理论中材料长度参数的区别、偶应力／应变梯度有限元法和无网格法的发展状况、偶应力／应变梯度理论的具体应用。
第二章总结了常见的几种偶应力／应变梯度理论,详细推导了这些理论的弹塑性本构关系,并分析了各理论之间的联系和区别。
第三章基于位移-转角不独立的偶应力理论,提供了微尺度下纯弯曲梁和悬臂梁的弹塑性解。通过与数值解对比,证实了这两种解的可靠性。结合相应的微弯曲实验,这两种解析解可用来标定细观偶应力／应变梯度理论的材料长度参数。
第四章总结了多种细观偶应力／应变梯度理论的刚塑性解,结合微扭转和微弯曲实验数据,标定出了各理论的材料长度参数。通过对比可以发现：材料长度参数严重地依赖于偶应力／应变梯度理论的类型。
第五章给出了两种偶应力／应变梯度有限元—八节点等参元和18参精化三角形单元,并将之应用于弹塑性领域。通过分析纯弯曲问题和孔边应力集中问题,发现采用罚函数有限元法计算位移-转角不独立的偶应力理论存在着问题。通过分析变形局部化问题,证实了偶应力／应变梯度理论可以有效地避免传统连续体力学存在的网格依赖性现象。
Classical continuum theory includes the homogenization assumption that the material properties remain the same from macroscale to mesoscale, which has been successfully applied in structural analysis on macroscale. However, when the size scales down to mesoscale or microscale, the effect of micro-defects in material (e.g., dislocation, micro-void, micro-inclusion) emerges, which results in inhomogeneous material properties and is represented as size effects in practice. In this case, classical continuum theory is not applicable. Moreover, the pathological mesh-dependency is exhibited when classical finite element method is applied in localization problems of softening materials. Therefore, other theories are needed to solve both size effects and mesh-dependency problems.
Based on Cosserat theory, many kinds of mesoscale couple stress/strain gradient theories have been developed. All of these theories include the material length parameters which can be used to describe the size of micro-defects and the width of localized deformation in softening material, and therefore can successfully predict the size effects and eliminate the mesh-dependency. However, there are some basic problems in mesoscale couple stress/strain gradient theories, e.g.,
(1) The inconsistent opinions on how to build continuum theory at the mesoscale or microscale lead to the fact that unlike in the elastic mechanics, there does not exist a standard framework for the current couple stress/strain gradient theories, which are built on different concepts. Consequently, lots of mesoscale couple stress/strain gradient theories are proposed, more than 10 of which are frequently cited. It increases the difficulties in choosing proper theory for engineering application.
(2) Besides several strain gradient theory based on physical mechanism, most theories include the material length parameters without specific physical meaning. The characteristics of these parameters need further research. At present, the micro-tests are used to determine the material length parameters in general, while exact and stable analytical solutions in couple stress/strain gradient elasto-plasticity are lacked as theoretical basis.
Focusing on the above problems, this paper compares the characteristics of various couple stress/strain gradient theories which can be taken as reference in engineering applications, analyzes the characteristics of material length parameters based on theoretical and numerical methods, and develops two kinds of micro-bend solutions in couple stress elasto-plasticity for the convenience of determining material length parameters. The thesis is arranged as follows:
In chapter 1, recent advances in mesoscale couple stress/strain gradient theories are introduced in detail, which includes the engineering backgrounds and characteristics of these theories, the differences of material length parameters between various theories, the corresponding finite element methods and mesh-free methods, and their specific applications.
In chapter 2, the author summarizes several couple stress/strain gradient theories and provides the corresponding elasto-plastic constitutive relations. By analyzing the relations and differences between these theories, some basic principles are proposed as references in choosing couple stress/strain gradient theory.
In chapter 3, the elasto-plastic solutions of pure-bending beam and cantilever beam are brought forward based on the couple stress theory where rotations depend on displacements. By comparison with numerical results, it is proved that both the solutions are reliable. With the corresponding micro-bend tests, both the analytical solutions can be used to determine material length parameters in mesoscale couple stress/strain gradient theories.
In chapter 4, the rigid-plastic solutions are summarized in various mesoscale couple stress/strain gradient theories. Combining the solutions with experimental results of micro-torsion and micro-bend, the material length parameters in various theories are determined. Finally, it is observed that these parameters are deeply dependent on the type of couple stress/strain gradient theory.
In chapter 5, two kinds of couple stress/strain gradient elements, i.e.,8-node parametric element and 18-DOF refined triangular element, are given and applied in elasto-plastic problems. By analyzing pure-bending and stress concentration problems, it is proved that the penalty approach is not stable in finite element implementation of the couple stress theory where rotations depend on displacements. By analyzing localization problems, it is shown that couple stress/strain gradient theories can effectively eliminate the mesh-dependency in classical continuum theory.
在Cosserat理论的基础上,众多学者发展出了多种细观偶应力／应变梯度理论。这些理论均引入了材料长度参数,可以衡量材料微缺陷的尺寸和软化材料局部化变形的宽度,因此可以成功地解决尺寸效应和网格依赖性问题。但是,细观偶应力／应变梯度理论存在着几个基本问题：
(1)如何构建微细观尺度的连续体理论没有统一的认识,目前的偶应力／应变梯度理论不像弹性力学那样拥有标准的理论框架,它们多是基于不同思想建立起来的。这导致了细观偶应力／应变梯度理论类型繁多,目前常见的理论有十多种,工程应用难以取舍；
(2)除了个别的基于细观机制的应变梯度理论,多数理论包含的材料长度参数没有明确的物理意义,它们的特点还有待进一步研究。目前常采用实验手段标定材料长度参数,但还缺乏精确稳定的偶应力／应变梯度弹塑性解析解作为标定的理论依据。
本文就这两个基本问题展开研究,详细对比了各种理论的特点以便作为工程使用的参考,采用理论推导和数值计算的手段分析了材料长度参数的特点,并构造了两种微弯曲偶应力弹塑性解析解作为标定材料长度参数的依据。各章的具体安排如下：
第一章详细给出了细观偶应力／应变梯度理论的发展近况,包括理论的工程背景、各理论的特点简介、各理论中材料长度参数的区别、偶应力／应变梯度有限元法和无网格法的发展状况、偶应力／应变梯度理论的具体应用。
第二章总结了常见的几种偶应力／应变梯度理论,详细推导了这些理论的弹塑性本构关系,并分析了各理论之间的联系和区别。
第三章基于位移-转角不独立的偶应力理论,提供了微尺度下纯弯曲梁和悬臂梁的弹塑性解。通过与数值解对比,证实了这两种解的可靠性。结合相应的微弯曲实验,这两种解析解可用来标定细观偶应力／应变梯度理论的材料长度参数。
第四章总结了多种细观偶应力／应变梯度理论的刚塑性解,结合微扭转和微弯曲实验数据,标定出了各理论的材料长度参数。通过对比可以发现：材料长度参数严重地依赖于偶应力／应变梯度理论的类型。
第五章给出了两种偶应力／应变梯度有限元—八节点等参元和18参精化三角形单元,并将之应用于弹塑性领域。通过分析纯弯曲问题和孔边应力集中问题,发现采用罚函数有限元法计算位移-转角不独立的偶应力理论存在着问题。通过分析变形局部化问题,证实了偶应力／应变梯度理论可以有效地避免传统连续体力学存在的网格依赖性现象。
Classical continuum theory includes the homogenization assumption that the material properties remain the same from macroscale to mesoscale, which has been successfully applied in structural analysis on macroscale. However, when the size scales down to mesoscale or microscale, the effect of micro-defects in material (e.g., dislocation, micro-void, micro-inclusion) emerges, which results in inhomogeneous material properties and is represented as size effects in practice. In this case, classical continuum theory is not applicable. Moreover, the pathological mesh-dependency is exhibited when classical finite element method is applied in localization problems of softening materials. Therefore, other theories are needed to solve both size effects and mesh-dependency problems.
Based on Cosserat theory, many kinds of mesoscale couple stress/strain gradient theories have been developed. All of these theories include the material length parameters which can be used to describe the size of micro-defects and the width of localized deformation in softening material, and therefore can successfully predict the size effects and eliminate the mesh-dependency. However, there are some basic problems in mesoscale couple stress/strain gradient theories, e.g.,
(1) The inconsistent opinions on how to build continuum theory at the mesoscale or microscale lead to the fact that unlike in the elastic mechanics, there does not exist a standard framework for the current couple stress/strain gradient theories, which are built on different concepts. Consequently, lots of mesoscale couple stress/strain gradient theories are proposed, more than 10 of which are frequently cited. It increases the difficulties in choosing proper theory for engineering application.
(2) Besides several strain gradient theory based on physical mechanism, most theories include the material length parameters without specific physical meaning. The characteristics of these parameters need further research. At present, the micro-tests are used to determine the material length parameters in general, while exact and stable analytical solutions in couple stress/strain gradient elasto-plasticity are lacked as theoretical basis.
Focusing on the above problems, this paper compares the characteristics of various couple stress/strain gradient theories which can be taken as reference in engineering applications, analyzes the characteristics of material length parameters based on theoretical and numerical methods, and develops two kinds of micro-bend solutions in couple stress elasto-plasticity for the convenience of determining material length parameters. The thesis is arranged as follows:
In chapter 1, recent advances in mesoscale couple stress/strain gradient theories are introduced in detail, which includes the engineering backgrounds and characteristics of these theories, the differences of material length parameters between various theories, the corresponding finite element methods and mesh-free methods, and their specific applications.
In chapter 2, the author summarizes several couple stress/strain gradient theories and provides the corresponding elasto-plastic constitutive relations. By analyzing the relations and differences between these theories, some basic principles are proposed as references in choosing couple stress/strain gradient theory.
In chapter 3, the elasto-plastic solutions of pure-bending beam and cantilever beam are brought forward based on the couple stress theory where rotations depend on displacements. By comparison with numerical results, it is proved that both the solutions are reliable. With the corresponding micro-bend tests, both the analytical solutions can be used to determine material length parameters in mesoscale couple stress/strain gradient theories.
In chapter 4, the rigid-plastic solutions are summarized in various mesoscale couple stress/strain gradient theories. Combining the solutions with experimental results of micro-torsion and micro-bend, the material length parameters in various theories are determined. Finally, it is observed that these parameters are deeply dependent on the type of couple stress/strain gradient theory.
In chapter 5, two kinds of couple stress/strain gradient elements, i.e.,8-node parametric element and 18-DOF refined triangular element, are given and applied in elasto-plastic problems. By analyzing pure-bending and stress concentration problems, it is proved that the penalty approach is not stable in finite element implementation of the couple stress theory where rotations depend on displacements. By analyzing localization problems, it is shown that couple stress/strain gradient theories can effectively eliminate the mesh-dependency in classical continuum theory.
引文
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