含孔洞压力敏感性材料裂纹尖端渐近场的研究
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摘要
对裂纹尖端渐近场的研究是断裂力学研究的重要课题之一。通过深入研究裂纹尖端物理量（应力、应变等）的分布及其力学本质，可以为建立材料的破坏准则以及评价结构的可靠性提供理论上的参考依据。
对裂纹尖端场做渐近分析是一个十分复杂的问题。为了简化问题的复杂性，在裂纹尖端渐近场的弹塑性分析中，人们通常假设材料是塑性不可压缩的。然而，自然界中的大量材料并不是这样的，它们在外载荷的作用下会产生较大的塑性体积变形，是塑性可压缩的，称之为压力敏感性材料（如：岩石、土壤、泡沫金属、聚合物、橡胶材料等）。在这类材料中通常含有复杂的微观结构（如：微裂纹、夹杂、孔洞等），影响着裂纹尖端渐近场物理量的分布，因而对含孔洞压力敏感性材料裂纹尖端渐近场进行研究更具普遍意义。
本文在综述裂纹尖端渐近场研究的历史和现状的基础上，采用损伤力学、断裂力学和弹塑性力学的理论，对含孔洞压力敏感性材料裂纹尖端渐近场问题进行深入细致的研究。主要工作如下：
1、运用Gurson模型，推导出含细观参数和压力敏感性材料参数的塑性宏观屈服面方程，讨论了基体材料参数和损伤参数（孔隙度）对宏观屈服面的影响，建立了线性硬化条件下含孔洞压力敏感性材料的本构方程。
2、运用压力敏感性材料的抛物型屈服准则和正交流动法则下的本构方程，在平面应力条件下，研究了压力敏感性材料起始扩展裂纹问题。给出I型、II型以及临界状态起始扩展裂纹尖端渐近场的基本构造为：I型裂纹可构造为两个弹性区和一个扇形区的“三区解”；II型裂纹可构造为两个弹性区、两个均匀应力区和三个扇形区的“七区解”；对于临界状态则较为复杂，当压力敏感性系数m=0时，可构造为两个均匀应力区和一个扇形区的“三区解”，当压力敏感性系数m>0时，可构造为一个均匀应力区、一个弹性区和一个扇形区的“三区解”。通过数值计算给出了裂纹尖端应力的角分布曲线，讨论了压力敏感性系数m对裂纹尖端渐近场的影响，讨论了场中应变的奇异性。
3、运用含孔洞压力敏感性材料的屈服准则和线性硬化条件下的本构方程，在平面应力条件下，研究了含孔洞压力敏感性材料准静态裂纹问题。根据奇异性量级分析，推导出含孔洞压力敏感性材料准静态裂纹尖端的塑性区和弹性区的控制方程。运用边界条件和裂纹特点计算出初值，采用双参数打靶法，给出了I型、II型裂纹尖端的渐进解。绘制了不同参数下的角分布曲线。讨论了不同参数对裂纹尖端应力场、速度场的影响。结果表明，孔隙度f对I型和II型裂纹尖端渐近场的影响都较大，压力敏感性系数m对I型裂纹尖端渐近场影响较大，对II型裂纹尖端渐近场影响较小。
4、运用含孔洞的压力敏感性材料的屈服准则和线性硬化条件下的本构方程，在平面应力条件下，研究了含孔洞压力敏感性材料在刚性表面上的界面裂纹问题。根据界面裂纹的特点及边界条件，计算出初值，采用双参数打靶法，给出了I型、II型裂纹尖端的渐进解。绘制了不同参数下的角分布曲线。讨论了不同参数对裂纹尖端应力场、速度场的影响。结果表明，材料常数αG对I型和II型裂纹尖端渐近场的影响较大，孔隙度f对I型裂纹尖端渐近场的影响较大，而对II型裂纹尖端渐近场的影响较小。
裂纹尖端渐近场的力学分析一直是一个十分复杂的力学问题。本文运用含孔洞压力敏感性材料的屈服准则和本构模型，给出了裂纹尖端的构造和控制方程，得出了渐近解，绘制了相应的角分布曲线，讨论了不同参数对裂纹尖端渐近场的影响。这些工作对裂纹尖端渐近场问题的研究进行了有益探索，为解决实际工程中的裂纹问题和建立材料的破坏准则提供理论参考依据，具有重要的现实意义。
Study on asymptotic crack tip field is an important subject of fracture mechanics. In orderto provide theoretical reference for the material failure criterion and the structural reliability, wemust in-depth study the distribution of the crack tip physical quantities (stress, strain, etc.) and itsmechanical essence.
Asymptotic analysis of the crack tip field is a complex problem. In order to reduce thecomplexity, we often assume that the material is plastic incompressible in the asymptotic cracktip field. However, a large number of materials called pressure sensitive material (such as rocks,soil, foam metal, polymer, rubber materials, etc.) will produce a large plastic volumetricdegeneration under external loads. These materials contain complex microscopic structures (suchas micro-cracks, inclusions, pores, etc.) which effect the distribution of physical parameters ofthe asymptotic crack tip field under external loads. Therefore, it is a universal significant to studyon asymptotic crack tip field in pressure sensitive material.
Based on the study of the history and current situation of the asymptotic crack tip field, thisdissertation researches the asymptotic crack tip field through the damage mechanics, fracturemechanics and elastic-plastic mechanics theory. The main work is summarized as follows:
1.From Gurson model, the plastic macroscopic yield surface equation containing themesoscopic parameters and pressure-sensitive parameters are obtained. The effect of themacroscopic yield surface equation with the matrix material parameters and injury parameters(porosity) are discussed. Under the linear hardening conditions, the constitutive equation forpressure-sensitive materials with holes is established.
2. Under plane stress condition, with the parabolic yield criterion and orthogonal flow rulethe constitutive equation for pressure sensitive materials is established. The basic structure ofmode I, mode II and the critical state of the asymptotic crack tip field before growing issummarized as follows: For mode I crack, it is divided into two elastic zones and a fan zone ofthe “solution with three zones”. For mode II crack, it is divided into two elastic zones, twouniform stress zones and three fan zones “solution with seven zones”. For the critical state, it isdivided into two uniform stress zones and a fan zone of the “solution with three zones” whenm=0. And it is divided into one uniform stress zones, one elastic zone and a fan zone of the“solution with three zones” when m>0. Angular variation curve of stress for asymptotic cracktip field is obtained. The effect of the pressure sensitive parameters m for asymptotic crack tip field is discussed. The singularity of the strain field is also discussed.
3. Under the plane stress conditions, quasi-static propagating crack for thepressure-sensitive materials with holes is studied. According to the analysis of singularity, theplastic zone and elastic zone control equation of the crack tip is deduced for the pressuresensitive material with holes. With the characteristics of the quasi-static propagating crack andboundary conditions, the initial value is calculated. Using the two-parameters shooting method,mode I and mode II asymptotic solutions in crack tip is obtained. Angular variation curve ofstress and velocity for asymptotic crack tip field is obtained. The impact of the crack tip stressfield and velocity field with the different parameters are discussed. The results show that theporosity f take great impact of the asymptotic crack tip field and the pressure sensitivitycoefficient m take greater impact of the mode I than mode on the asymptotic crack tip field.
4. Under the plane stress conditions, rigid interface crack propagation for thepressure-sensitive materials with holes is studied. According to the characteristics of theinterface crack and boundary conditions, the initial value is calculated. Using the two-parametersshooting method, mode I and mode II asymptotic solutions in crack tip is obtained. Angularvariation curve of stress and velocity for asymptotic crack tip field is obtained. The impact of thecrack tip stress field and velocity field with the different parameters are discussed. The resultsshow that the porosity f take great impact of the asymptotic crack tip field and the materialconstantsα Gtake greater impact of the mode I than mode on the asymptotic crack tip field.
The mechanical analysis of the asymptotic crack tip field is a very complex problem. In thisdissertation, the pressure sensitive materials yield condition and the constitutive model is used.The structure and the control equation of the crack tip deduced and the asymptotic solution isobtained. The angular distribution curve is drawn and the impact of the asymptotic crack tip fieldwith different parameters discussed. This work is useful of the asymptotic crack tip field. It hasimportant significance for the actual project and the failure criterion.
对裂纹尖端场做渐近分析是一个十分复杂的问题。为了简化问题的复杂性，在裂纹尖端渐近场的弹塑性分析中，人们通常假设材料是塑性不可压缩的。然而，自然界中的大量材料并不是这样的，它们在外载荷的作用下会产生较大的塑性体积变形，是塑性可压缩的，称之为压力敏感性材料（如：岩石、土壤、泡沫金属、聚合物、橡胶材料等）。在这类材料中通常含有复杂的微观结构（如：微裂纹、夹杂、孔洞等），影响着裂纹尖端渐近场物理量的分布，因而对含孔洞压力敏感性材料裂纹尖端渐近场进行研究更具普遍意义。
本文在综述裂纹尖端渐近场研究的历史和现状的基础上，采用损伤力学、断裂力学和弹塑性力学的理论，对含孔洞压力敏感性材料裂纹尖端渐近场问题进行深入细致的研究。主要工作如下：
1、运用Gurson模型，推导出含细观参数和压力敏感性材料参数的塑性宏观屈服面方程，讨论了基体材料参数和损伤参数（孔隙度）对宏观屈服面的影响，建立了线性硬化条件下含孔洞压力敏感性材料的本构方程。
2、运用压力敏感性材料的抛物型屈服准则和正交流动法则下的本构方程，在平面应力条件下，研究了压力敏感性材料起始扩展裂纹问题。给出I型、II型以及临界状态起始扩展裂纹尖端渐近场的基本构造为：I型裂纹可构造为两个弹性区和一个扇形区的“三区解”；II型裂纹可构造为两个弹性区、两个均匀应力区和三个扇形区的“七区解”；对于临界状态则较为复杂，当压力敏感性系数m=0时，可构造为两个均匀应力区和一个扇形区的“三区解”，当压力敏感性系数m>0时，可构造为一个均匀应力区、一个弹性区和一个扇形区的“三区解”。通过数值计算给出了裂纹尖端应力的角分布曲线，讨论了压力敏感性系数m对裂纹尖端渐近场的影响，讨论了场中应变的奇异性。
3、运用含孔洞压力敏感性材料的屈服准则和线性硬化条件下的本构方程，在平面应力条件下，研究了含孔洞压力敏感性材料准静态裂纹问题。根据奇异性量级分析，推导出含孔洞压力敏感性材料准静态裂纹尖端的塑性区和弹性区的控制方程。运用边界条件和裂纹特点计算出初值，采用双参数打靶法，给出了I型、II型裂纹尖端的渐进解。绘制了不同参数下的角分布曲线。讨论了不同参数对裂纹尖端应力场、速度场的影响。结果表明，孔隙度f对I型和II型裂纹尖端渐近场的影响都较大，压力敏感性系数m对I型裂纹尖端渐近场影响较大，对II型裂纹尖端渐近场影响较小。
4、运用含孔洞的压力敏感性材料的屈服准则和线性硬化条件下的本构方程，在平面应力条件下，研究了含孔洞压力敏感性材料在刚性表面上的界面裂纹问题。根据界面裂纹的特点及边界条件，计算出初值，采用双参数打靶法，给出了I型、II型裂纹尖端的渐进解。绘制了不同参数下的角分布曲线。讨论了不同参数对裂纹尖端应力场、速度场的影响。结果表明，材料常数αG对I型和II型裂纹尖端渐近场的影响较大，孔隙度f对I型裂纹尖端渐近场的影响较大，而对II型裂纹尖端渐近场的影响较小。
裂纹尖端渐近场的力学分析一直是一个十分复杂的力学问题。本文运用含孔洞压力敏感性材料的屈服准则和本构模型，给出了裂纹尖端的构造和控制方程，得出了渐近解，绘制了相应的角分布曲线，讨论了不同参数对裂纹尖端渐近场的影响。这些工作对裂纹尖端渐近场问题的研究进行了有益探索，为解决实际工程中的裂纹问题和建立材料的破坏准则提供理论参考依据，具有重要的现实意义。
Study on asymptotic crack tip field is an important subject of fracture mechanics. In orderto provide theoretical reference for the material failure criterion and the structural reliability, wemust in-depth study the distribution of the crack tip physical quantities (stress, strain, etc.) and itsmechanical essence.
Asymptotic analysis of the crack tip field is a complex problem. In order to reduce thecomplexity, we often assume that the material is plastic incompressible in the asymptotic cracktip field. However, a large number of materials called pressure sensitive material (such as rocks,soil, foam metal, polymer, rubber materials, etc.) will produce a large plastic volumetricdegeneration under external loads. These materials contain complex microscopic structures (suchas micro-cracks, inclusions, pores, etc.) which effect the distribution of physical parameters ofthe asymptotic crack tip field under external loads. Therefore, it is a universal significant to studyon asymptotic crack tip field in pressure sensitive material.
Based on the study of the history and current situation of the asymptotic crack tip field, thisdissertation researches the asymptotic crack tip field through the damage mechanics, fracturemechanics and elastic-plastic mechanics theory. The main work is summarized as follows:
1.From Gurson model, the plastic macroscopic yield surface equation containing themesoscopic parameters and pressure-sensitive parameters are obtained. The effect of themacroscopic yield surface equation with the matrix material parameters and injury parameters(porosity) are discussed. Under the linear hardening conditions, the constitutive equation forpressure-sensitive materials with holes is established.
2. Under plane stress condition, with the parabolic yield criterion and orthogonal flow rulethe constitutive equation for pressure sensitive materials is established. The basic structure ofmode I, mode II and the critical state of the asymptotic crack tip field before growing issummarized as follows: For mode I crack, it is divided into two elastic zones and a fan zone ofthe “solution with three zones”. For mode II crack, it is divided into two elastic zones, twouniform stress zones and three fan zones “solution with seven zones”. For the critical state, it isdivided into two uniform stress zones and a fan zone of the “solution with three zones” whenm=0. And it is divided into one uniform stress zones, one elastic zone and a fan zone of the“solution with three zones” when m>0. Angular variation curve of stress for asymptotic cracktip field is obtained. The effect of the pressure sensitive parameters m for asymptotic crack tip field is discussed. The singularity of the strain field is also discussed.
3. Under the plane stress conditions, quasi-static propagating crack for thepressure-sensitive materials with holes is studied. According to the analysis of singularity, theplastic zone and elastic zone control equation of the crack tip is deduced for the pressuresensitive material with holes. With the characteristics of the quasi-static propagating crack andboundary conditions, the initial value is calculated. Using the two-parameters shooting method,mode I and mode II asymptotic solutions in crack tip is obtained. Angular variation curve ofstress and velocity for asymptotic crack tip field is obtained. The impact of the crack tip stressfield and velocity field with the different parameters are discussed. The results show that theporosity f take great impact of the asymptotic crack tip field and the pressure sensitivitycoefficient m take greater impact of the mode I than mode on the asymptotic crack tip field.
4. Under the plane stress conditions, rigid interface crack propagation for thepressure-sensitive materials with holes is studied. According to the characteristics of theinterface crack and boundary conditions, the initial value is calculated. Using the two-parametersshooting method, mode I and mode II asymptotic solutions in crack tip is obtained. Angularvariation curve of stress and velocity for asymptotic crack tip field is obtained. The impact of thecrack tip stress field and velocity field with the different parameters are discussed. The resultsshow that the porosity f take great impact of the asymptotic crack tip field and the materialconstantsα Gtake greater impact of the mode I than mode on the asymptotic crack tip field.
The mechanical analysis of the asymptotic crack tip field is a very complex problem. In thisdissertation, the pressure sensitive materials yield condition and the constitutive model is used.The structure and the control equation of the crack tip deduced and the asymptotic solution isobtained. The angular distribution curve is drawn and the impact of the asymptotic crack tip fieldwith different parameters discussed. This work is useful of the asymptotic crack tip field. It hasimportant significance for the actual project and the failure criterion.
引文
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