基于辛空间的解析奇异单元及其在断裂力学中的应用
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摘要
众所周知,裂纹问题普遍存在于工程结构中,如飞行器、火箭、轮船、锅炉、桥梁等,它所引起的破坏事故往往会造成巨大的损失,因此断裂力学的研究对预防和控制裂纹引起的事故具有重大的实际意义。断裂力学中通常采用应力强度因子来衡量裂纹尖端场的强弱,并使用相应的应力强度因子准则来判断裂纹是否会发生失稳扩展。同时,在一些基于内聚力模型的裂纹问题中, COD准则被用作判断裂纹是否向前扩展的依据。因此,准确有效的计算出含裂纹结构的应力强度因子以及COD等参数是工程实用结构强度分析十分关心的课题。
虽然断裂力学基础理论部分的研究已经得到比较完善的发展,但仍然存在一些复杂的情况并没有得到很好的解决,如裂纹表面受任意荷载的情况、由多种各向异性材料组成的多材料裂纹情况等。这主要是由于在传统求解体系下相应的求解步骤过于复杂,导致理论求解无法实现。应用力学辛求解体系的提出使得很多以往无法求解的复杂问题得到解析求解,这其中也包括断裂问题。本文首先在辛求解体系下研究了一些复杂的裂纹问题,并求解出相应的解析辛本征解和特解。然后,基于这些解析解和特解,构建了一系列奇异单元,分别用于处理不同情况下的含裂纹结构问题的分析。本文主要的工作内容如下：
(1)在平面极坐标辛求解体系下,给出了平面单材料、双材料裂纹表面受任意荷载所对应的特解。通过将荷载近似的展开为多项式叠加的形式,解析求解出每一个展开项所对应的特解,从而可以得到任意荷载作用下所对应的特解。同时,建立了反平面裂纹问题的辛求解体系,并给出了反平面裂纹表面受任意荷载情况下的解析辛本征解和特解。通过坐标转换的手段,简化了多材料反平面裂纹问题的分析过程,并提供了几个具体问题的应力奇异性特征方程。此外,本文还研究了在普通辛体系下无法直接解决的由多种各向异性材料组成的反平面裂纹问题。通过将子域法与辛方法相结合,提出了一种可以近似求解出反平面裂纹尖端应力奇异性阶数的方法,并通过数值算例验证了本方法的有效性。
(2)基于辛体系所提供的平面单材料、双材料裂纹以及反平面裂纹的解析辛本征解以及特解,构造了一系列解析奇异单元。通过在裂纹尖端使用奇异元,而在结构的其它区域使用常规单元,可以对含裂纹结构进行有效的数值分析,并通过展开项系数与应力强度因子之间的关系,不需要借助任何后处理手段,可以直接高精度地给出Ⅰ,Ⅱ以及Ⅲ型应力强度因子,从而保证了求解的精度。同时,证明了奇异元的刚度阵与其半径大小无关,这一特点保证了数值计算的稳定性。作为一个尝试,本文还将所构建的奇异单元用于疲劳裂纹扩展问题的分析,从而可以得到更加精细的裂纹扩展路径以及更加准确的结构疲劳寿命的估计值。数值算例表明,所构建的奇异元具有非常好的数值求解精度和稳定性,是含裂纹结构分析的一种非常有效的数值方法。
(3)基于辛体系所给出的平面裂纹问题的解析辛本征解以及特解,构造了用于分析单材料和双材料Ⅰ+Ⅱ混合型Dugdale裂纹问题的奇异单元。它与常规单元相结合,可有效用于混合型Dugdale模型单材料、双材料裂纹问题的数值分析,并通过迭代可以得到包括塑性区长度,裂纹尖端张开/滑开位移,虚拟裂纹上作用的内聚力大小在内的全部参数。同时,证明了奇异元的刚度阵与其半径大小无关,这一特点保证了数值计算的稳定性。此外,该奇异元还可以应用到双材料含桥联力Ⅰ型Dugdale裂纹问题的分析。数值算例验证了本文方法具有精度高,迭代收敛速度快的优点。所提出的解析奇异单元可有效应用于弹塑性材料含裂纹问题的数值分析,并具有明显的实用价值。
本文所构建的系列奇异单元有效提高了含裂纹结构问题求解的精度和效率,并且具有非常好的数值稳定性。同时,它不需要过渡单元,具有较强的通用性以及兼容性,可以直接集成到大多数已有结构有限元程序系统中,充分发挥其在实际工程结构分析中的应用价值。
It's well known that fracture problems arise frequently in many engineering structures, such as in aircrafts, rockets, ships, boilers, bridges, etc. The study on fracture mechanics including prevention and control has significant meaning since destroy accidents caused by fractures could always bring huge loses. Stress intensity factors are usually employed to judge the strength of stress and displacement fields around crack tips, and the corresponding strength criterion of instability propagation was proposed. Meanwhile, COD criterions are used in the crack problems based on cohesive models. Hence, calculating the parameters such as stress intensity factors and COD accurately and efficiently are very attractive research subjects in the strength analysis of engineering practicle structures.
The basic theoretical study of fracture problems is well developed, but there are still some complex cases left unsolved, such as cracks with arbitrary tractions, anisotropic multi-material cracks, etc. The solving procedure for such complex problems in traditional methods is always too complex to achieve. With the presence of symplectic dual approach for applied mechanics, many complex problems previously considered impossible can be solved analytically, such as crack problems. In this study, some complex crack problems are considered in symplectic solving system, and analytical symplectic eigen solutions and special solutions are specified. Then, based on these analytical solutions and special solutions, a series of singular finite element are constructed for the numerical analysis of different crack problems. The main contents of the present study are expressed as follows:
(1). The special solutions for arbitrary crack tractions acting on single material and bimaterial cracks under plane assumptions are specified in symplectic solving system. Crack tractions are expressed approximately in terms of polynomial expansion, and special solution of each expanding term is specified analytically. In this way, the final special solutions for arbitrary crack tractions can be obtained accordingly. Simultaneously, the symplectic solving system for antiplane crack problem is proposed, analytical symplectic eigen solution and special solution for antiplane cracks with arbitrary tractions are specified. The analysis procedure of multi-material antiplane crack problem is simplified using the coordinate transform technique. Some specific cases are considered as an example, and the corresponding eigen equation of stress singularity orders are given. Besides, multi-material anisotropic antiplane crack problem which couldn't be solved by traditional sympectic approach is considered in this study. By combining subfield method and symplectic method, a new approximate approach for the stress singularity analyze of multi-material anisotropic antiplane crack is proposed, and this new approach is illustrated by numerical examples.
(2). A series of singular finite element are constructed, respectively, based on analytical symplectic eigen solutions and special solutions for single material, bimaterial cracks under plane assumptions and those of antiplane cracks. Using the singular element around crack tips and conventional elements in the other area, cracked structures can be analyzed numerically with high efficiency. According to the relationship between expanding coefficients and stress intensity factors, stress intensity factors for mode Ⅰ,Ⅱ and Ⅲ can be obtained directly without any post process, hence the computational accuracy is ensured. Simultaneously, stiffness matrix of the singular finite elements are proved to be independent on element sizes, this feature is helpful to improve the stability of numerical calculation. As an attempt, the singular finite element for plane cracks is further applied on fatigue crack growth in order to get more precise growth path and more accurate estimated fatigue life. Numerical examples imply that the singular finite element which possesses good caltulation accuracy and stability is very effective for numerical analysis of cracked structures.
(3). Based on analytical symplectic eigne solution and special solutions of plane cracks obtained using symplectic system, singular finite elements respectively for single material and bimaterial Ⅰ+Ⅱ mixed-mode Dugdale model based cracks are constructed. Using the singular element around crack tips and conventional elements in the other area, mixed-mode Dugdale crack problems can be solved numerically. Simultaneously, length of plastic zone, crack tip opening/sliding displacement, cohesive stress acting on virtual crack and other parameters in mixed-mode Dugdale model can all be specified by iteration. The stiffness matrix is proved to be independent on element size, and this feature of the singular finite element is helpful to ensure the stability of calculation. Furthermore, the singular element is applied on analyze of mode Ⅰ bimaterial Dugdale cracks with bridging tractions. Numerical examples imply that the present method possesses high calculation accuracy and fast iteration speed, etc. The singular element presented in this study has significant practical value, and it can be applied on the numerical analysis of cracks in elastro-plastic materials with high efficiency.
A series of singular finite element are constructed, and the new method which has good stability of calculation has impoved the calculation accuracy and efficiency of numerical analysis of cracked structures. Also, transition elements are not required which has improved the commonality and compatibility of the present elements, hence, the singular finite elements can be integrated into most existing FEM softwares directly to make the most of its practical value in the analysis of engineering structures.
虽然断裂力学基础理论部分的研究已经得到比较完善的发展,但仍然存在一些复杂的情况并没有得到很好的解决,如裂纹表面受任意荷载的情况、由多种各向异性材料组成的多材料裂纹情况等。这主要是由于在传统求解体系下相应的求解步骤过于复杂,导致理论求解无法实现。应用力学辛求解体系的提出使得很多以往无法求解的复杂问题得到解析求解,这其中也包括断裂问题。本文首先在辛求解体系下研究了一些复杂的裂纹问题,并求解出相应的解析辛本征解和特解。然后,基于这些解析解和特解,构建了一系列奇异单元,分别用于处理不同情况下的含裂纹结构问题的分析。本文主要的工作内容如下：
(1)在平面极坐标辛求解体系下,给出了平面单材料、双材料裂纹表面受任意荷载所对应的特解。通过将荷载近似的展开为多项式叠加的形式,解析求解出每一个展开项所对应的特解,从而可以得到任意荷载作用下所对应的特解。同时,建立了反平面裂纹问题的辛求解体系,并给出了反平面裂纹表面受任意荷载情况下的解析辛本征解和特解。通过坐标转换的手段,简化了多材料反平面裂纹问题的分析过程,并提供了几个具体问题的应力奇异性特征方程。此外,本文还研究了在普通辛体系下无法直接解决的由多种各向异性材料组成的反平面裂纹问题。通过将子域法与辛方法相结合,提出了一种可以近似求解出反平面裂纹尖端应力奇异性阶数的方法,并通过数值算例验证了本方法的有效性。
(2)基于辛体系所提供的平面单材料、双材料裂纹以及反平面裂纹的解析辛本征解以及特解,构造了一系列解析奇异单元。通过在裂纹尖端使用奇异元,而在结构的其它区域使用常规单元,可以对含裂纹结构进行有效的数值分析,并通过展开项系数与应力强度因子之间的关系,不需要借助任何后处理手段,可以直接高精度地给出Ⅰ,Ⅱ以及Ⅲ型应力强度因子,从而保证了求解的精度。同时,证明了奇异元的刚度阵与其半径大小无关,这一特点保证了数值计算的稳定性。作为一个尝试,本文还将所构建的奇异单元用于疲劳裂纹扩展问题的分析,从而可以得到更加精细的裂纹扩展路径以及更加准确的结构疲劳寿命的估计值。数值算例表明,所构建的奇异元具有非常好的数值求解精度和稳定性,是含裂纹结构分析的一种非常有效的数值方法。
(3)基于辛体系所给出的平面裂纹问题的解析辛本征解以及特解,构造了用于分析单材料和双材料Ⅰ+Ⅱ混合型Dugdale裂纹问题的奇异单元。它与常规单元相结合,可有效用于混合型Dugdale模型单材料、双材料裂纹问题的数值分析,并通过迭代可以得到包括塑性区长度,裂纹尖端张开/滑开位移,虚拟裂纹上作用的内聚力大小在内的全部参数。同时,证明了奇异元的刚度阵与其半径大小无关,这一特点保证了数值计算的稳定性。此外,该奇异元还可以应用到双材料含桥联力Ⅰ型Dugdale裂纹问题的分析。数值算例验证了本文方法具有精度高,迭代收敛速度快的优点。所提出的解析奇异单元可有效应用于弹塑性材料含裂纹问题的数值分析,并具有明显的实用价值。
本文所构建的系列奇异单元有效提高了含裂纹结构问题求解的精度和效率,并且具有非常好的数值稳定性。同时,它不需要过渡单元,具有较强的通用性以及兼容性,可以直接集成到大多数已有结构有限元程序系统中,充分发挥其在实际工程结构分析中的应用价值。
It's well known that fracture problems arise frequently in many engineering structures, such as in aircrafts, rockets, ships, boilers, bridges, etc. The study on fracture mechanics including prevention and control has significant meaning since destroy accidents caused by fractures could always bring huge loses. Stress intensity factors are usually employed to judge the strength of stress and displacement fields around crack tips, and the corresponding strength criterion of instability propagation was proposed. Meanwhile, COD criterions are used in the crack problems based on cohesive models. Hence, calculating the parameters such as stress intensity factors and COD accurately and efficiently are very attractive research subjects in the strength analysis of engineering practicle structures.
The basic theoretical study of fracture problems is well developed, but there are still some complex cases left unsolved, such as cracks with arbitrary tractions, anisotropic multi-material cracks, etc. The solving procedure for such complex problems in traditional methods is always too complex to achieve. With the presence of symplectic dual approach for applied mechanics, many complex problems previously considered impossible can be solved analytically, such as crack problems. In this study, some complex crack problems are considered in symplectic solving system, and analytical symplectic eigen solutions and special solutions are specified. Then, based on these analytical solutions and special solutions, a series of singular finite element are constructed for the numerical analysis of different crack problems. The main contents of the present study are expressed as follows:
(1). The special solutions for arbitrary crack tractions acting on single material and bimaterial cracks under plane assumptions are specified in symplectic solving system. Crack tractions are expressed approximately in terms of polynomial expansion, and special solution of each expanding term is specified analytically. In this way, the final special solutions for arbitrary crack tractions can be obtained accordingly. Simultaneously, the symplectic solving system for antiplane crack problem is proposed, analytical symplectic eigen solution and special solution for antiplane cracks with arbitrary tractions are specified. The analysis procedure of multi-material antiplane crack problem is simplified using the coordinate transform technique. Some specific cases are considered as an example, and the corresponding eigen equation of stress singularity orders are given. Besides, multi-material anisotropic antiplane crack problem which couldn't be solved by traditional sympectic approach is considered in this study. By combining subfield method and symplectic method, a new approximate approach for the stress singularity analyze of multi-material anisotropic antiplane crack is proposed, and this new approach is illustrated by numerical examples.
(2). A series of singular finite element are constructed, respectively, based on analytical symplectic eigen solutions and special solutions for single material, bimaterial cracks under plane assumptions and those of antiplane cracks. Using the singular element around crack tips and conventional elements in the other area, cracked structures can be analyzed numerically with high efficiency. According to the relationship between expanding coefficients and stress intensity factors, stress intensity factors for mode Ⅰ,Ⅱ and Ⅲ can be obtained directly without any post process, hence the computational accuracy is ensured. Simultaneously, stiffness matrix of the singular finite elements are proved to be independent on element sizes, this feature is helpful to improve the stability of numerical calculation. As an attempt, the singular finite element for plane cracks is further applied on fatigue crack growth in order to get more precise growth path and more accurate estimated fatigue life. Numerical examples imply that the singular finite element which possesses good caltulation accuracy and stability is very effective for numerical analysis of cracked structures.
(3). Based on analytical symplectic eigne solution and special solutions of plane cracks obtained using symplectic system, singular finite elements respectively for single material and bimaterial Ⅰ+Ⅱ mixed-mode Dugdale model based cracks are constructed. Using the singular element around crack tips and conventional elements in the other area, mixed-mode Dugdale crack problems can be solved numerically. Simultaneously, length of plastic zone, crack tip opening/sliding displacement, cohesive stress acting on virtual crack and other parameters in mixed-mode Dugdale model can all be specified by iteration. The stiffness matrix is proved to be independent on element size, and this feature of the singular finite element is helpful to ensure the stability of calculation. Furthermore, the singular element is applied on analyze of mode Ⅰ bimaterial Dugdale cracks with bridging tractions. Numerical examples imply that the present method possesses high calculation accuracy and fast iteration speed, etc. The singular element presented in this study has significant practical value, and it can be applied on the numerical analysis of cracks in elastro-plastic materials with high efficiency.
A series of singular finite element are constructed, and the new method which has good stability of calculation has impoved the calculation accuracy and efficiency of numerical analysis of cracked structures. Also, transition elements are not required which has improved the commonality and compatibility of the present elements, hence, the singular finite elements can be integrated into most existing FEM softwares directly to make the most of its practical value in the analysis of engineering structures.
引文
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