基于J-积分和构型力理论的材料断裂行为研究
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摘要
传统的连续介质力学(也称变形力体系)主要研究连续体的空间运动及其变形。近年来,连续介质力学一个新的分支,构型力理论(或者叫材料空间力学理论)逐渐发展起来以处理与连续体的结构演化与缺陷运动有关的力学问题。本文旨在开展热力学框架下的构型力理论研究,并将其与传统的变形力体系相结合来分析断裂问题。这不但为断裂力学研究提供新的思路和方法,而且也有助于深化对断裂问题的认识。以下是本文的主要研究内容、思路、方法和结论:
第一章回顾和评述了宏观断裂力学的分析方法,包括以奇异场理论和围道积分为基础的经典断裂力学、构型力方法以及内聚力模型。
第二章从传统的变形力体系出发研究了增量塑性本构关系下的断裂问题,重点分析了裂纹扩展过程中的能量耗散与裂纹扩展阻力。对于包含尖劈裂纹的弹塑性体,总的能量耗散包括塑性区耗散与裂尖耗散;通过裂纹扩展的局部稳态特性,裂尖耗散可以表示成无限小围道上的J_(tip)~(ep)-积分。与Rice的J -积分相比, J~(ep)-积分具有以下特征:其被积函数的能量项为自由能密度而非应力功密度;无论对增量塑性还是形变塑性, J~(ep)-积分始终是路径相关的;在裂纹稳态扩展条件下,用于评价裂纹扩展阻力的远场J_(far)~(ep)-积分可以写成裂尖耗散与塑性耗散之和,这个结果为J_(far)~(ep)-积分作为裂纹扩展阻力评价指标提供了一定的物理基础。从理论分析和数值模拟两方面评价了J_(far)~(ep)-积分裂纹扩展阻力和能量耗散率扩展阻力Γ(Δa )。计算结果表明, J_(far)~(ep)-积分裂纹扩展阻力随着裂纹扩展而增加,能量耗散率扩展阻力Γ(Δa )随裂纹扩展而减小,只有当裂纹扩展趋于稳态条件时,两者才逐渐趋于一致。采用Gurson损伤模型模拟了弹塑性材料钝化裂纹扩展,分析了材料硬化指数、孔洞体积含量和弹性极限应变对裂纹扩展阻力和断裂功的影响,结果表明断裂功与应力三轴度相关,而非固定的材料常数。
第三章引入了构型力和演化控制体概念,建立了热/机械载荷与力/电载荷作用下的构型力基本方程。对于演化控制体,外界所做的功除了传统的变形功外还包括由于控制体内的物质增减引起的构型功。给出了物质观察者的坐标转换关系,通过所有的外力功(包括变形功与构型功)在物质观察者发生任意刚体运动的情况下保持不变这一基本原理建立了多物理场下构型力(矩)平衡。通过热力学第一定律以及构型功与演化控制体边界切向演化速度无关这一内在要求导出了Eshelby关系,即构型力与变形力的关系;对于热机械过程,通过演化控制体的热力学第二定律确定了构型热。
第四章采用构型力理论分析了热弹性材料与电弹性材料的断裂问题。通过含缺陷的演化控制体的热力学第二定律推导了裂纹尖端的能量耗散,结果表明,无论在热/机械载荷还是力/电载荷作用下,裂尖处负的集中内部构型体积力在裂纹扩展方向的投影正是裂纹扩展驱动力。这表明裂纹扩展由裂尖处集中的内部构型体积力来控制,其做的功并不转化为自由能而是对应着不可逆过程的耗散。通过含裂尖演化控制体的广义动能定理导出了惯性集中构型力,进一步通过裂尖平衡方程求得内部集中构型力。
除了奇异断裂理论以外,内聚力模型也广泛应用于各类材料与结构的断裂问题中。在论文的第五章,提出了界面变形梯度的概念,利用构型力理论的基本思想构建了材料空间应力矢量和界面分离位移矢量,在此基础上建立了材料空间下的不可逆内聚力模型,该模型能同时满足客观性原理和断裂能的加载路径相关性要求。随后,从变分原理出发推导出了含内聚区变形体的有限元离散形式,结合ABAQUS有限元软件,给出了内聚力单元用户子程序(UEL)。最后,模拟了双悬臂梁界面裂纹问题,分析了加载构型与界面性能对试件的极限承载能力的影响。
The traditional continuum mechanics which is rooted on deformational forces system mainly deals with the problems of the spacial movement and deformation of continuum. Recently, a new branch of continuum mechanics, configurational force which is also called material space mechanics, is developed to represent the evolution of material structures and the motion of defects. The present thesis primarily aims at carrying out the study of configurational forces theory in the framework of thermodynamics, and on this basis the fracture problems are investigated by combining configurational forces approach and the deformational forces system. This not only provides a fresh look and method onto the classical fracture mechanics, but also deepens the understanding on fracture problems. The main ideas, methods and conclusions are summarized as following:
In Chapter 1, the methods dealing with fracture problems are reviewed, including the classical fracture mechanics theory characterized by singular field theory and contour integrals, configurational forces approach to fracture mechanics and cohesive zone model.
In Chapter 2, starting from the deformational forces system, the fracture problem of materials with the incremental plasticity response is investigated with emphasis on the calculation of the energy dissipation and the evaluation of crack growth resistance during crack propagation. For the elastic-plastic body containing sharp crack, the total energy dissipation consists of the energy dissipation in plastic zone and that concentrated at the crack tip. Using local steady-state condition of crack propagation, crack tip dissipation can be reformulated as the form of J teipp-integral which is the value of J ep-integral evaluated along the infinite small contour circling the crack tip. Compared with Rice’s J -integral, J ep-integral has the following particularities: the energy-momentum tensor in J ep-integral is defined by the free energy density rather than stress working density; J~(ep)-integral is path-dependent no matter incremental plasticity or deformation plasticity; J_(farss)~(ep) -integral,the value of J~(ep)-integral evaluated along the far boundary contour under the global steady-state condition of crack propagation, can be rewritten as the sum of J~(tip)~(ep)-integral and the plastic dissipation, which indicates that J_(far)~(ep)-integral can be taken as a candidate to evaluate crack growth resistance due to the reasonable physical meaning. Both the J_(far)~(ep)-integral crack growth resistance and energy dissipation rate resistance are evaluated theoretically and numerically. Simulation results show that J_(far)~(ep)-integral resistance increases with the crack growth, whereas energy dissipation rate resistance decreases. When the steady-state condition is reached approximately the two resistances lie in the same level. At last, the crack propagation is simulated by using Gurson model. The effects of material hardening level, the void volume fraction and the yield strain on fracture energy and crack growth resistances are discussed and the numerical results demonstrate that fracture energy is stress-triaxiality-dependent instead of a material constant.
In Chapter 3, the concepts of configurational forces and migrating control volume are introduced and the basic equations of configurational forces are established for thermomechanical and electro-mechanical loading using Gurtin’s ideas. The working expended on migrating control volume consists of not only deformation working but also configurational working which accounts for the addition and deletion of material particles through the boundary of the migrating control volume. The exchange of material observer is given. The balance equations of configurational forces and moment of multi-physical fields are established by using the basic principle that the total working on migrating control volume holds invariant under arbitrary translation and rigid rotation of material observer. Eshelby relation relating deformational forces and configurational forces is identified by combining the first thermodynamic law and the requirement that configurational working is independent of the tangential component of evolution velocity of the boundary of the migrating control volume. The configurational heating of thermomechanical process can be determined from the second thermodynamic law.
In Chapter 4, the configurational forces are used to investigate the fracture problems of thermoelastic and electroelastic materials. Energy dissipation concentrated at the crack tip is derived from the generalized mechanical version of the second law of thermodynamics applicable to migrating control volume. Theoretical derivation shows that the negative projection of the internal configurational force concentrated at the crack tip along the direction of crack propagation plays the role of energy release rate no matter thermomechanical process or electro-mechanical processe. This indicates that the internal concentrated configurational force acts to govern crack propagation and its working is converted into irreversible energy dissipation. Furthermore, the inertia part of the concentrated configurational force at the crack tip is determined from the generalized kinetic energy theorem. Further, the internal configurational force concentrated at the crack tip can be obtained by substituting the inertia part of the configurational force into the configurational force balance.
In addition to singular field theory, cohesive zone model is also widely used to study the fracture problems of materials and components. In Chapter 5, the concept of interfacial deformation gradient is introduced and the traction and separation displacement vectors of material space are constructed by using the basic ideas of configurational forces. On this basis, the irreversible cohesive zone model of material space is developed. The cohesive law not only meets the requirement of objectivity principle but also predicts the path-dependent cohesive energy. The discrete form of balance equation of the continuum containing cohesive zone are derived using variational principle and the finite element solution is implemented by coding user element subroutine (UEL) of the cohesive zone model in the commercial software ABAQUS. At last, the interfacial crack propagation of double cantilever beam (DCB) is simulated, and the effects of interfacial parameters and loading configuration on the loading capacity of DCB specimen are discussed.
第一章回顾和评述了宏观断裂力学的分析方法,包括以奇异场理论和围道积分为基础的经典断裂力学、构型力方法以及内聚力模型。
第二章从传统的变形力体系出发研究了增量塑性本构关系下的断裂问题,重点分析了裂纹扩展过程中的能量耗散与裂纹扩展阻力。对于包含尖劈裂纹的弹塑性体,总的能量耗散包括塑性区耗散与裂尖耗散;通过裂纹扩展的局部稳态特性,裂尖耗散可以表示成无限小围道上的J_(tip)~(ep)-积分。与Rice的J -积分相比, J~(ep)-积分具有以下特征:其被积函数的能量项为自由能密度而非应力功密度;无论对增量塑性还是形变塑性, J~(ep)-积分始终是路径相关的;在裂纹稳态扩展条件下,用于评价裂纹扩展阻力的远场J_(far)~(ep)-积分可以写成裂尖耗散与塑性耗散之和,这个结果为J_(far)~(ep)-积分作为裂纹扩展阻力评价指标提供了一定的物理基础。从理论分析和数值模拟两方面评价了J_(far)~(ep)-积分裂纹扩展阻力和能量耗散率扩展阻力Γ(Δa )。计算结果表明, J_(far)~(ep)-积分裂纹扩展阻力随着裂纹扩展而增加,能量耗散率扩展阻力Γ(Δa )随裂纹扩展而减小,只有当裂纹扩展趋于稳态条件时,两者才逐渐趋于一致。采用Gurson损伤模型模拟了弹塑性材料钝化裂纹扩展,分析了材料硬化指数、孔洞体积含量和弹性极限应变对裂纹扩展阻力和断裂功的影响,结果表明断裂功与应力三轴度相关,而非固定的材料常数。
第三章引入了构型力和演化控制体概念,建立了热/机械载荷与力/电载荷作用下的构型力基本方程。对于演化控制体,外界所做的功除了传统的变形功外还包括由于控制体内的物质增减引起的构型功。给出了物质观察者的坐标转换关系,通过所有的外力功(包括变形功与构型功)在物质观察者发生任意刚体运动的情况下保持不变这一基本原理建立了多物理场下构型力(矩)平衡。通过热力学第一定律以及构型功与演化控制体边界切向演化速度无关这一内在要求导出了Eshelby关系,即构型力与变形力的关系;对于热机械过程,通过演化控制体的热力学第二定律确定了构型热。
第四章采用构型力理论分析了热弹性材料与电弹性材料的断裂问题。通过含缺陷的演化控制体的热力学第二定律推导了裂纹尖端的能量耗散,结果表明,无论在热/机械载荷还是力/电载荷作用下,裂尖处负的集中内部构型体积力在裂纹扩展方向的投影正是裂纹扩展驱动力。这表明裂纹扩展由裂尖处集中的内部构型体积力来控制,其做的功并不转化为自由能而是对应着不可逆过程的耗散。通过含裂尖演化控制体的广义动能定理导出了惯性集中构型力,进一步通过裂尖平衡方程求得内部集中构型力。
除了奇异断裂理论以外,内聚力模型也广泛应用于各类材料与结构的断裂问题中。在论文的第五章,提出了界面变形梯度的概念,利用构型力理论的基本思想构建了材料空间应力矢量和界面分离位移矢量,在此基础上建立了材料空间下的不可逆内聚力模型,该模型能同时满足客观性原理和断裂能的加载路径相关性要求。随后,从变分原理出发推导出了含内聚区变形体的有限元离散形式,结合ABAQUS有限元软件,给出了内聚力单元用户子程序(UEL)。最后,模拟了双悬臂梁界面裂纹问题,分析了加载构型与界面性能对试件的极限承载能力的影响。
The traditional continuum mechanics which is rooted on deformational forces system mainly deals with the problems of the spacial movement and deformation of continuum. Recently, a new branch of continuum mechanics, configurational force which is also called material space mechanics, is developed to represent the evolution of material structures and the motion of defects. The present thesis primarily aims at carrying out the study of configurational forces theory in the framework of thermodynamics, and on this basis the fracture problems are investigated by combining configurational forces approach and the deformational forces system. This not only provides a fresh look and method onto the classical fracture mechanics, but also deepens the understanding on fracture problems. The main ideas, methods and conclusions are summarized as following:
In Chapter 1, the methods dealing with fracture problems are reviewed, including the classical fracture mechanics theory characterized by singular field theory and contour integrals, configurational forces approach to fracture mechanics and cohesive zone model.
In Chapter 2, starting from the deformational forces system, the fracture problem of materials with the incremental plasticity response is investigated with emphasis on the calculation of the energy dissipation and the evaluation of crack growth resistance during crack propagation. For the elastic-plastic body containing sharp crack, the total energy dissipation consists of the energy dissipation in plastic zone and that concentrated at the crack tip. Using local steady-state condition of crack propagation, crack tip dissipation can be reformulated as the form of J teipp-integral which is the value of J ep-integral evaluated along the infinite small contour circling the crack tip. Compared with Rice’s J -integral, J ep-integral has the following particularities: the energy-momentum tensor in J ep-integral is defined by the free energy density rather than stress working density; J~(ep)-integral is path-dependent no matter incremental plasticity or deformation plasticity; J_(farss)~(ep) -integral,the value of J~(ep)-integral evaluated along the far boundary contour under the global steady-state condition of crack propagation, can be rewritten as the sum of J~(tip)~(ep)-integral and the plastic dissipation, which indicates that J_(far)~(ep)-integral can be taken as a candidate to evaluate crack growth resistance due to the reasonable physical meaning. Both the J_(far)~(ep)-integral crack growth resistance and energy dissipation rate resistance are evaluated theoretically and numerically. Simulation results show that J_(far)~(ep)-integral resistance increases with the crack growth, whereas energy dissipation rate resistance decreases. When the steady-state condition is reached approximately the two resistances lie in the same level. At last, the crack propagation is simulated by using Gurson model. The effects of material hardening level, the void volume fraction and the yield strain on fracture energy and crack growth resistances are discussed and the numerical results demonstrate that fracture energy is stress-triaxiality-dependent instead of a material constant.
In Chapter 3, the concepts of configurational forces and migrating control volume are introduced and the basic equations of configurational forces are established for thermomechanical and electro-mechanical loading using Gurtin’s ideas. The working expended on migrating control volume consists of not only deformation working but also configurational working which accounts for the addition and deletion of material particles through the boundary of the migrating control volume. The exchange of material observer is given. The balance equations of configurational forces and moment of multi-physical fields are established by using the basic principle that the total working on migrating control volume holds invariant under arbitrary translation and rigid rotation of material observer. Eshelby relation relating deformational forces and configurational forces is identified by combining the first thermodynamic law and the requirement that configurational working is independent of the tangential component of evolution velocity of the boundary of the migrating control volume. The configurational heating of thermomechanical process can be determined from the second thermodynamic law.
In Chapter 4, the configurational forces are used to investigate the fracture problems of thermoelastic and electroelastic materials. Energy dissipation concentrated at the crack tip is derived from the generalized mechanical version of the second law of thermodynamics applicable to migrating control volume. Theoretical derivation shows that the negative projection of the internal configurational force concentrated at the crack tip along the direction of crack propagation plays the role of energy release rate no matter thermomechanical process or electro-mechanical processe. This indicates that the internal concentrated configurational force acts to govern crack propagation and its working is converted into irreversible energy dissipation. Furthermore, the inertia part of the concentrated configurational force at the crack tip is determined from the generalized kinetic energy theorem. Further, the internal configurational force concentrated at the crack tip can be obtained by substituting the inertia part of the configurational force into the configurational force balance.
In addition to singular field theory, cohesive zone model is also widely used to study the fracture problems of materials and components. In Chapter 5, the concept of interfacial deformation gradient is introduced and the traction and separation displacement vectors of material space are constructed by using the basic ideas of configurational forces. On this basis, the irreversible cohesive zone model of material space is developed. The cohesive law not only meets the requirement of objectivity principle but also predicts the path-dependent cohesive energy. The discrete form of balance equation of the continuum containing cohesive zone are derived using variational principle and the finite element solution is implemented by coding user element subroutine (UEL) of the cohesive zone model in the commercial software ABAQUS. At last, the interfacial crack propagation of double cantilever beam (DCB) is simulated, and the effects of interfacial parameters and loading configuration on the loading capacity of DCB specimen are discussed.
引文
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