含复杂界面非均匀材料断裂力学研究
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摘要
复合材料已经在越来越多的领域中得到应用。尽管人们将许多复合材料设计成宏观上属性连续变化的非均匀材料,但各种复合材料都会或多或少地存在材料界面,尤其是颗粒增强复合材料。颗粒增强材料能够明显改善材料的刚度、强度和耐磨性,但其断裂韧性有时明显地低于基体材料。复合材料的服役环境一般都比较苛刻,断裂破坏是其最常见的失效模式之一。当考察复合材料的断裂性能时,我们不得不面对复合材料内部的材料界面。为此,本文将对复杂界面环境中裂纹的力学行为进行研究。
第1章首先回顾了颗粒增强复合材料断裂问题的研究现状,接着介绍了扩展有限元方法,最后总结了当前用于求解裂纹尖端断裂参数的数值计算方法,包括求解裂纹尖端的应力强度因子和T应力。由于扩展有限元方法允许裂纹面或材料界面独立于有限元网格,所以该方法可以方便地求解含复杂界面材料的静态断裂问题和裂纹扩展问题。然而,当裂纹尖端附近存在复杂的材料界面时,目前却没有一种方法能够方便、准确地提取裂纹尖端的断裂参数。因此,本文的研究目的是给出一个在裂纹尖端附近存在复杂材料界面情况下能够容易提取裂纹尖端断裂参数的方法。
第2章推导了用于求解裂纹尖端应力强度因子的相互作用积分,得到了一个新的区域积分表达式。相互作用积分基于一个包含两个相容力学场(真实场和辅助场)的守恒积分。我们对相互作用积分做了两项改进:第一,通过定义一个恰当的辅助场,可以发现相互作用积分中的材料导数项消失了。第二,证明了积分区域内的材料界面并不影响相互作用积分的有效性。因此,本章得到的相互作用积分可以方便地求解属性连续或不连续变化的材料内部裂纹尖端的应力强度因子。将相互作用积分方法与扩展有限元方法相结合,求解了一些典型断裂问题,很好地验证了相互作用积分的有效性和区域无关性。然后,通过选择四种材料属性来考察材料连续性对混合型应力强度因子的影响。数值结果显示,材料属性及其一阶导数的连续性对I型和II型应力强度因子影响很大,而材料属性的高阶导数对应力强度因子的影响不大。
实际上,裂纹可能在单一材料中扩展,也可能沿材料界面向前扩展。因此,第3章考察了界面裂纹问题。首先,求解了两个非均匀材料界面间的界面裂纹尖端应力奇异性。然后,推导了求解界面裂纹尖端应力强度因子的相互作用积分。与第2章中的相互作用积分相似,本章得到的相互作用积分的区域积分形式也不含有材料导数,并且也不受积分区域内部其它材料界面的影响。因此,本章得到的相互作用积分可以求解含复杂界面的材料内部界面裂纹尖端的应力强度因子。结合扩展有限元法,验证了相互作用积分方法的可靠性和积分区域无关性。最后,我们考察了几个典型的非均匀材料的界面断裂问题。
与二维断裂问题相比,三维断裂问题无疑更具有工程实际意义。第4章考察了用于求解三维曲线裂纹前沿应力强度因子的相互作用积分方法,导出了一个不含有材料属性导数项的三维区域积分表达式,该表达式也允许积分区域内部的材料属性不连续。将其与有限元法结合计算了典型的三维断裂问题,将结果与已发表文章对比,两者吻合很好,这说明相互作用积分方法能够有效求解三维裂纹尖端的应力强度因子。相互作用积分的区域无关性在算例中也得到了很好的验证。
除了应力强度因子之外,T应力也是一个重要的断裂控制参数。为此,第5章通过选择作用于裂纹尖端的集中力所引起的力学场作为辅助场,获得了求解裂纹尖端T应力的相互作用积分。它与求解裂纹尖端应力强度因子的相互作用积分相同,也具有积分不含材料导数项和不要求积分区域内材料属性连续的优点。接着,我们对相互作用积分求解T应力的可行性给出了严格证明。通过计算典型的断裂问题,相互作用积分求解T应力的有效性和区域无关性得到了数值验证。最后,我们考察了材料连续性对裂纹尖端T应力的影响。结果显示,材料属性及其一阶导数的连续性对T应力影响非常大,而其高阶导数的连续性对T应力无明显影响。
Composite materials have been applied in more and more fields. Although composite materials have been designed with continuous and nonhomogeneous properties in macro scale, there are more or less material interfaces in various composite materials, especially, in particle reinforced composite materials (PRCMs). It is often found that although PRCMs can significantly improve the strength, stiffness and wear resistance of materials, the fracture toughness is significantly lower than that of the matrix material. Since composite materials are usually used in severe conditions, fracture is one of the most common failure modes. The material interfaces have to be taken into account when the fracture performance of these composites is concerned. Therefore, the mechanical behaviors of a crack in the environment containing complex interfaces are investigated in this thesis.
In Chapter 1, the fracture problems of PRCMs are reviewed firstly. Then, the extended finite element method (XFEM) is introduced. Finally, the numerical methods are described for extracting fracture parameters, including the stress intensity factors (SIFs) and the T-stress. Since the XFEM allows cracks or material interfaces to be independent of the mesh, it can be used to deal with static crack problems and crack propagation problems of the materials with complex interfaces conveniently. However, up to the present day, there is no method which can not extract fracture parameters exactly and conveniently for the crack surrounded by complex interfaces. Accordingly, the aim of this article is to develop a method for extracting the fracture parameters easily when the crack tips lie in the vicinity of complex interfaces.
In Chapter 2, a new domain expression of the interaction integral is derived for the computation of mixed-mode SIFs. This method is based on a conservation integral that relies on two admissible mechanical states (actual and auxiliary fields). Two improvements are provided for the interaction integral. First, by a suitable definition of the auxiliary fields, it is found that in the interaction integral, the terms related to the derivatives of material properties vanish. Second, we provide the proof that the formulation is still valid even when the integral domain contains material interfaces. Therefore, the interaction integral derived here can be used to solve the SIFs of a crack in nonhomogeneous materials with continuous or discontinuous properties. The interaction integral method combined with the XFEM is used to solve several representative fracture problems to verify the validation and domain-independence of the interaction integral. Then, the influences of material continuity on the mixed-mode SIFs are investigated by selecting four types of material properties. Numerical results show that the mechanical properties and their first-order derivatives affect mode I and II SIFs greatly, while the higher-order derivatives affect the SIFs slightly.
In practice, a crack may grow in one material or along the material interface. Therefore, in Chapter 3, the interface crack problems are investigated. At the beginning of Chapter 3, the stress singularity of the interface crack between two nonhomogeneous materials is solved. Then, an interaction integral is derived for obtaining mixed-mode SIFs of an interface crack. Similarly to the expression in Chapter 2, the domain integral form of the interaction integral does not contain any derivatives of material properties and is valid when there are other material interfaces in the integral domain. Thus, the derived formulation can be applied to deal with interfacial fracture problems of the materials with complex interfaces. The interaction integral combined with the XFEM is employed to solve some fracture problems and the results show that the method is very reliable and domain-independent. Finally, several representative examples of complicated interface crack problems between nonhomogeneous materials are considered.
There is no doubt that three-dimensional (3D) fracture problems are more significant in engineering fields compared with two-dimensional (2D) crack problems. In Chapter 4, the interaction integral for solving mixed-mode SIFs along a 3D curved crack front is discussed. A new 3D domain formulation without containing any derivatives of material properties is obtained. The interaction integral is still valid when the material properties in the integral domain are discontinuous. This method in conjunction with the finite element method (FEM) is employed to solve several representative 3D fracture problems. According to the comparison between the results and those from the published lectures, good agreement demonstrates the validation of the interaction integral. The domain-independence of the interaction integral is also shown in the results.
Except for the SIFs, the T-stress is also an important fracture parameter. Therefore, in Chapter 5, the method for extracting the T-stress is described. Selecting the auxiliary field which is caused by a centralized force at the crack tip, we derived a new domain expression of the interaction integral for the computation of the T-stress. The interaction integral for extracting the T-stress has the same advantage as that for solving the SIFs, i.e., the interaction integral does not contain the terms related to the derivatives of material properties and does not require the material properties in the integral domain to be continuous. Then, the feasibility to use the interaction integral to extract the T-stress is proved rigorously. The interaction integral shows good validation and domain-independence by solving several representative fracture problems. Finally, the influences of material continuity on the T-stress are investigated. It can be found that the mechanical properties and their first-order derivatives affect the T-stress greatly, while the higher-order derivatives affect the T-stress slightly.
第1章首先回顾了颗粒增强复合材料断裂问题的研究现状,接着介绍了扩展有限元方法,最后总结了当前用于求解裂纹尖端断裂参数的数值计算方法,包括求解裂纹尖端的应力强度因子和T应力。由于扩展有限元方法允许裂纹面或材料界面独立于有限元网格,所以该方法可以方便地求解含复杂界面材料的静态断裂问题和裂纹扩展问题。然而,当裂纹尖端附近存在复杂的材料界面时,目前却没有一种方法能够方便、准确地提取裂纹尖端的断裂参数。因此,本文的研究目的是给出一个在裂纹尖端附近存在复杂材料界面情况下能够容易提取裂纹尖端断裂参数的方法。
第2章推导了用于求解裂纹尖端应力强度因子的相互作用积分,得到了一个新的区域积分表达式。相互作用积分基于一个包含两个相容力学场(真实场和辅助场)的守恒积分。我们对相互作用积分做了两项改进:第一,通过定义一个恰当的辅助场,可以发现相互作用积分中的材料导数项消失了。第二,证明了积分区域内的材料界面并不影响相互作用积分的有效性。因此,本章得到的相互作用积分可以方便地求解属性连续或不连续变化的材料内部裂纹尖端的应力强度因子。将相互作用积分方法与扩展有限元方法相结合,求解了一些典型断裂问题,很好地验证了相互作用积分的有效性和区域无关性。然后,通过选择四种材料属性来考察材料连续性对混合型应力强度因子的影响。数值结果显示,材料属性及其一阶导数的连续性对I型和II型应力强度因子影响很大,而材料属性的高阶导数对应力强度因子的影响不大。
实际上,裂纹可能在单一材料中扩展,也可能沿材料界面向前扩展。因此,第3章考察了界面裂纹问题。首先,求解了两个非均匀材料界面间的界面裂纹尖端应力奇异性。然后,推导了求解界面裂纹尖端应力强度因子的相互作用积分。与第2章中的相互作用积分相似,本章得到的相互作用积分的区域积分形式也不含有材料导数,并且也不受积分区域内部其它材料界面的影响。因此,本章得到的相互作用积分可以求解含复杂界面的材料内部界面裂纹尖端的应力强度因子。结合扩展有限元法,验证了相互作用积分方法的可靠性和积分区域无关性。最后,我们考察了几个典型的非均匀材料的界面断裂问题。
与二维断裂问题相比,三维断裂问题无疑更具有工程实际意义。第4章考察了用于求解三维曲线裂纹前沿应力强度因子的相互作用积分方法,导出了一个不含有材料属性导数项的三维区域积分表达式,该表达式也允许积分区域内部的材料属性不连续。将其与有限元法结合计算了典型的三维断裂问题,将结果与已发表文章对比,两者吻合很好,这说明相互作用积分方法能够有效求解三维裂纹尖端的应力强度因子。相互作用积分的区域无关性在算例中也得到了很好的验证。
除了应力强度因子之外,T应力也是一个重要的断裂控制参数。为此,第5章通过选择作用于裂纹尖端的集中力所引起的力学场作为辅助场,获得了求解裂纹尖端T应力的相互作用积分。它与求解裂纹尖端应力强度因子的相互作用积分相同,也具有积分不含材料导数项和不要求积分区域内材料属性连续的优点。接着,我们对相互作用积分求解T应力的可行性给出了严格证明。通过计算典型的断裂问题,相互作用积分求解T应力的有效性和区域无关性得到了数值验证。最后,我们考察了材料连续性对裂纹尖端T应力的影响。结果显示,材料属性及其一阶导数的连续性对T应力影响非常大,而其高阶导数的连续性对T应力无明显影响。
Composite materials have been applied in more and more fields. Although composite materials have been designed with continuous and nonhomogeneous properties in macro scale, there are more or less material interfaces in various composite materials, especially, in particle reinforced composite materials (PRCMs). It is often found that although PRCMs can significantly improve the strength, stiffness and wear resistance of materials, the fracture toughness is significantly lower than that of the matrix material. Since composite materials are usually used in severe conditions, fracture is one of the most common failure modes. The material interfaces have to be taken into account when the fracture performance of these composites is concerned. Therefore, the mechanical behaviors of a crack in the environment containing complex interfaces are investigated in this thesis.
In Chapter 1, the fracture problems of PRCMs are reviewed firstly. Then, the extended finite element method (XFEM) is introduced. Finally, the numerical methods are described for extracting fracture parameters, including the stress intensity factors (SIFs) and the T-stress. Since the XFEM allows cracks or material interfaces to be independent of the mesh, it can be used to deal with static crack problems and crack propagation problems of the materials with complex interfaces conveniently. However, up to the present day, there is no method which can not extract fracture parameters exactly and conveniently for the crack surrounded by complex interfaces. Accordingly, the aim of this article is to develop a method for extracting the fracture parameters easily when the crack tips lie in the vicinity of complex interfaces.
In Chapter 2, a new domain expression of the interaction integral is derived for the computation of mixed-mode SIFs. This method is based on a conservation integral that relies on two admissible mechanical states (actual and auxiliary fields). Two improvements are provided for the interaction integral. First, by a suitable definition of the auxiliary fields, it is found that in the interaction integral, the terms related to the derivatives of material properties vanish. Second, we provide the proof that the formulation is still valid even when the integral domain contains material interfaces. Therefore, the interaction integral derived here can be used to solve the SIFs of a crack in nonhomogeneous materials with continuous or discontinuous properties. The interaction integral method combined with the XFEM is used to solve several representative fracture problems to verify the validation and domain-independence of the interaction integral. Then, the influences of material continuity on the mixed-mode SIFs are investigated by selecting four types of material properties. Numerical results show that the mechanical properties and their first-order derivatives affect mode I and II SIFs greatly, while the higher-order derivatives affect the SIFs slightly.
In practice, a crack may grow in one material or along the material interface. Therefore, in Chapter 3, the interface crack problems are investigated. At the beginning of Chapter 3, the stress singularity of the interface crack between two nonhomogeneous materials is solved. Then, an interaction integral is derived for obtaining mixed-mode SIFs of an interface crack. Similarly to the expression in Chapter 2, the domain integral form of the interaction integral does not contain any derivatives of material properties and is valid when there are other material interfaces in the integral domain. Thus, the derived formulation can be applied to deal with interfacial fracture problems of the materials with complex interfaces. The interaction integral combined with the XFEM is employed to solve some fracture problems and the results show that the method is very reliable and domain-independent. Finally, several representative examples of complicated interface crack problems between nonhomogeneous materials are considered.
There is no doubt that three-dimensional (3D) fracture problems are more significant in engineering fields compared with two-dimensional (2D) crack problems. In Chapter 4, the interaction integral for solving mixed-mode SIFs along a 3D curved crack front is discussed. A new 3D domain formulation without containing any derivatives of material properties is obtained. The interaction integral is still valid when the material properties in the integral domain are discontinuous. This method in conjunction with the finite element method (FEM) is employed to solve several representative 3D fracture problems. According to the comparison between the results and those from the published lectures, good agreement demonstrates the validation of the interaction integral. The domain-independence of the interaction integral is also shown in the results.
Except for the SIFs, the T-stress is also an important fracture parameter. Therefore, in Chapter 5, the method for extracting the T-stress is described. Selecting the auxiliary field which is caused by a centralized force at the crack tip, we derived a new domain expression of the interaction integral for the computation of the T-stress. The interaction integral for extracting the T-stress has the same advantage as that for solving the SIFs, i.e., the interaction integral does not contain the terms related to the derivatives of material properties and does not require the material properties in the integral domain to be continuous. Then, the feasibility to use the interaction integral to extract the T-stress is proved rigorously. The interaction integral shows good validation and domain-independence by solving several representative fracture problems. Finally, the influences of material continuity on the T-stress are investigated. It can be found that the mechanical properties and their first-order derivatives affect the T-stress greatly, while the higher-order derivatives affect the T-stress slightly.
引文
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