韧性材料损伤的微尺度效应及其机理
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摘要
近年来,微机电系统技术快速发展,各种微器件、微机械的主导尺寸常常处在微米或亚微米量级。新近的一系列力学实验观察、理论和数值分析表明:在微米或亚微米量级下,材料的力学行为呈现出强烈的微尺度效应。基于传统尺度无关本构框架的结构强度分析方法和安全评价体系面临挑战。因此。对材料和结构中损伤的微尺度效应开展深入、系统的研究,不仅具有重要的理论意义,而且具有重要的工程实用价值。
本文运用应变梯度理论分析和动态离散位错模拟两种方法分别对微尺度下韧性材料损伤的微尺度效应及其机理进行了较深入、系统的研究。主要工作有: 1、从应变梯度理论和体胞模型出发,建立了尺度相关的含孔洞材料的塑性势,并将著名的Gurson模型推广到微尺度范围。通过对含理想球形孔洞的轴对称代表性胞元的分析,结果发现:(1)在孔洞体积分数一定的情况下,孔洞长大的速率低于尺度无关的Gurson模型预测的长大速率;(2)随着孔洞半径的减少,孔洞材料的屈服迹线逐渐向外扩张,显示出明显的尺度效应;在此基础上,通过数值积分给出了孔洞材料尺度相关的塑性势。
2、基于SG应变梯度形变理论和含孔洞的无限大体模型,研究了三轴应力场中孔洞形状效应和尺度效应对其长大的耦合作用。结果表明:(1)孔洞的长大是尺度相关的,且存在一个与远场应力三维度和孔洞形状无关的“孔洞临界半径”,当微孔洞的等效半径接近或小于临界半径时,孔洞难以通过塑性变形长大;(2)孔洞长大的尺度效应和形状效应是相互耦合的,一般地,孔洞形状越偏离理想球形,其长大的尺度效应越明显,此外,这种耦合与远程应力三维度密切相关。
3、基于SG应变梯度理论,研究了颗粒增强复合材料中颗粒的形状和尺寸对其内部及界面应力分布的影响,结果有助于理解金属基复合材料的微尺度增强机理。通过对三轴应力下含椭球夹杂的无限大体边值问题的分析,结果表明:(1)在微米尺度下,夹杂尺寸越小,夹杂/基体界面的法向应力、剪切应力和夹杂内部主拉伸方向的应力越高。(2)这种尺度效应随着远程应力三轴度的减小和远场等效应变的增加变得更为显著。(3)为更加合适的考虑基体/夹杂界面上的偶拽力平衡条件,本文引入了与尺度相关的界面能的概念,并进一步分析了界面能对颗粒界面及其内部应力分布的影响。
4、开发了基于离散位错动力学模拟的计算分析程序,研究了FCC单晶体内含孤立孔洞的长大机理,结果有助于揭示不同大小孔洞长大机制的内在差异。通过对平面应变等轴拉伸载荷下无限大单晶体内孔洞周围的离散位错模拟,结果表明:(1)位错剪切环从位错源形核后,位错环扩展并到达孔洞表面是单晶内孔洞长大的重要机制;(2)不同尺寸的孔洞呈现出不同的长大方式,当孔洞较大时,孔洞的长大随等效应变光滑变化,但当孔洞较小时,孔洞长大接近于线弹性的,且呈现出“蛙跳”式长大,造成这种差异的主要原因是,在不同大小的孔洞附近,位错的可动性和位错源的可激活性不同;(3)在微米尺度下,特别是孔洞半径足够小时,孔洞长大的呈现出明显的离散性,这些现象难以被高阶连续理论所捕捉。
5、基于位错发射的Rice-Thomson模型,研究了不同大小、不同取向的椭圆孔洞表面的位错发射及由此导致的孔洞长大机制。通过对单晶体中椭圆孔洞表面位错发射的分析得到:(1)存在一个临界应力,当外载低于该临界应力时,孔洞表面基本不发生位错发射,孔洞难以长大,当外载大于该临界应力时,孔洞表面发射位错并引起孔洞的突发性长大;(2)在纳米量级下,椭圆孔洞位错发射的临界应力呈现出尺度效应,孔洞的等效半径越小,位错发射所需临界应力就越大,而在同一等效半径下,椭圆孔洞的位错发射临界应力较圆孔洞小得多;(3)位错发射及由此引起的孔洞长大与椭圆孔洞主轴的取向有很大关系。
6、基于二维离散位错动力学模拟方法,研究了无限大FCC单晶体内微夹杂周围的应力场分布,讨论了夹杂处孔洞形核的可能机制。通过对等轴拉伸载荷下含微夹杂的FCC单晶体的离散位错模拟得到:(1)由于基体/夹杂界面附近产生的位错障碍和位错塞积,基体/夹杂界面上出现一系列的应力峰;(2)随着夹杂半径的减小,界面上应力峰的数目和应力峰值都不断减小。这些结果表明,微尺度下,孔洞在基体/夹杂界面形核可能与界面上应力峰的数目和峰值的大小有关,并且夹杂越小,孔洞形核越难,这与现有实验结果定性一致。
With the rapid development of Micro Electromechanical System (i.e. MEMS) in recent years, various micro-apparatus and micro-machines have been widely used. A series of micro-mechanical tests and analytic results have already indicated that the mechanical behaviors show strong scale effect when the sample falls within the micron size range. At the micron scale, the strength analyses and structure optimization based on the classical independent-scale constitutive relation usually fail to be efficient. Thus, deep and systemic researches on damage and deformation behavior at the micron scale are of fundamental importance not only in understanding the damage mechanism of solids but also in engineering application.
In the present dissertation, the micro-scale effect on damage and deformation behavior is investigated by both the strain gradient theory and the discrete dislocation dynamic modeling. The results from two kinds of methods are complementary with each other, which introduce more information about the subject to us. The main results are shown as follows:
Based on the SG strain gradient deformation theory and the infinite model with an isolated spheroidal void, the size and shape effects on void growth in triaxial stress field are carefully investigated. It is found that (1) there exists a“critical void radius”, the void is difficult to grow by plastic flow in the matrix material when the equivalent radius of micro-void approach or is lower than the“critical radius”; (2) the“critical radius”is independent of the void shape and the remote stress statue, and approximately equals to the material intrinsic length associated with the stretch strain gradient; (3)the void size effect and shape effect is coupled with each other, generally, the shape effect can enhance the scale effect. In addition, this coupling is closely related to the remote stress triaxiality.
Based on the strain gradient deformation theory and the representative volume element (RVE) model, a scale-dependent macroscopic plastic potential of porous material is deduced, which generalizes application of the classical Gurson model to the micron size range. Further analyses show that(1)at a given void volume fraction, the growth rate of micron sized void is lower than that predicated by the traditional R-T model and Gurson model; (2) the yield locus of porous materials expands outwards with the void size decreasing.
The effects of particle size and shape on the stress distribution at the particle/matrix interface and within the particle are investigated by employing the scale-dependent strain gradient SG theory to provide a better understanding to the micro-particle reinforcement mechanism. By solving a axial-symmetrical boundary value problem consisting of a infinite solid and an isolated spheriodal particle embedded in an infinite matrix, some interesting results are found as follow: (1) at the micron scale, the smaller the particle is, the higher the stresses at the matrix/inclusion interface and within the particle are; (2) this size effect is more remarkable with decreasing the stress triaxiality and increasing the remote equivalent strain. To reasonably equilibrate the double stress traction at the particle-matrix interface, a new parameter named“interfacial energy density”is introduced. The influence of“interfacial energy density”on the stress distribution is also investigated.
A program for discrete dislocation dynamic modeling is developed. Using this program, the growth of an isolated void embedded in an infinite FCC single crystal is investigated. The emphasis is focused on the size effect of void growth and its intrinsic mechanism. The calculation results show that: (1) the expansion of dislocation shear loop and reaching the void surface is main mechanism controlling the void growth in single crystal; (2) the void growth vs. the remote strain curves are continuous and smooth for the larger voids, but there are some steps on the curves for the smaller void, so the internal mechanism controlling the void growth is different for the voids with various sizes; (3)this size effect is closely associated with the dislocation mobility and the number of dislocation sources activated around the void; (4) the discreteness of void growth is inherent and significant for the micron sized void.
Based on the well-known Rice-Thomson model, dislocation emission and void growth induced by it are considered for elliptic voids with various sizes and different directions. The results indicate that: (1) there exist a“critical stress”, when the applied stress is lower than this“critical stress”, dislocations can not be emitted from the void surface, while when the applied stress is larger than the“critical stress”, the dislocation emission is activated and can introduce the nano-void to grow suddenly; (2) within the nanometer size range, the dislocation emission“critical stress”for elliptic void is size dependent, i.e. the smaller the void is, the higher the critical stress is, and at a given equivalent radius, the critical stress for the elliptic voids is lower than that for the circular voids; (3) the dislocation emission and the void growth induced by it are closely related to the direction angle between the slip plane and the axis of the elliptic void.
By employing the 2D discrete dislocation dynamic modeling, the stress distribution in the vicinity of an isolate circular particle embedded in an infinite FCC matrix is investigated, and the void nucleation mechanism is discussed. The results are shown that: (1) due to the dislocation obstruction and pile-ups near the interface, there are a series of peaks for the interfacial normal stress at the particle-matrix interface; (2) the number of interfacial stress peak and the magnitude of these peaks increases with increasing the inclusion radius. These results indicate that, within the micron scale range, the void nucleation at the matrix/particle interface seems to be related to the number and magnitude of the interfacial stress peak. According to this assumption, some experiment observations can be explained, i.e. the smaller the particle is, the more difficult the void nucleation is.
本文运用应变梯度理论分析和动态离散位错模拟两种方法分别对微尺度下韧性材料损伤的微尺度效应及其机理进行了较深入、系统的研究。主要工作有: 1、从应变梯度理论和体胞模型出发,建立了尺度相关的含孔洞材料的塑性势,并将著名的Gurson模型推广到微尺度范围。通过对含理想球形孔洞的轴对称代表性胞元的分析,结果发现:(1)在孔洞体积分数一定的情况下,孔洞长大的速率低于尺度无关的Gurson模型预测的长大速率;(2)随着孔洞半径的减少,孔洞材料的屈服迹线逐渐向外扩张,显示出明显的尺度效应;在此基础上,通过数值积分给出了孔洞材料尺度相关的塑性势。
2、基于SG应变梯度形变理论和含孔洞的无限大体模型,研究了三轴应力场中孔洞形状效应和尺度效应对其长大的耦合作用。结果表明:(1)孔洞的长大是尺度相关的,且存在一个与远场应力三维度和孔洞形状无关的“孔洞临界半径”,当微孔洞的等效半径接近或小于临界半径时,孔洞难以通过塑性变形长大;(2)孔洞长大的尺度效应和形状效应是相互耦合的,一般地,孔洞形状越偏离理想球形,其长大的尺度效应越明显,此外,这种耦合与远程应力三维度密切相关。
3、基于SG应变梯度理论,研究了颗粒增强复合材料中颗粒的形状和尺寸对其内部及界面应力分布的影响,结果有助于理解金属基复合材料的微尺度增强机理。通过对三轴应力下含椭球夹杂的无限大体边值问题的分析,结果表明:(1)在微米尺度下,夹杂尺寸越小,夹杂/基体界面的法向应力、剪切应力和夹杂内部主拉伸方向的应力越高。(2)这种尺度效应随着远程应力三轴度的减小和远场等效应变的增加变得更为显著。(3)为更加合适的考虑基体/夹杂界面上的偶拽力平衡条件,本文引入了与尺度相关的界面能的概念,并进一步分析了界面能对颗粒界面及其内部应力分布的影响。
4、开发了基于离散位错动力学模拟的计算分析程序,研究了FCC单晶体内含孤立孔洞的长大机理,结果有助于揭示不同大小孔洞长大机制的内在差异。通过对平面应变等轴拉伸载荷下无限大单晶体内孔洞周围的离散位错模拟,结果表明:(1)位错剪切环从位错源形核后,位错环扩展并到达孔洞表面是单晶内孔洞长大的重要机制;(2)不同尺寸的孔洞呈现出不同的长大方式,当孔洞较大时,孔洞的长大随等效应变光滑变化,但当孔洞较小时,孔洞长大接近于线弹性的,且呈现出“蛙跳”式长大,造成这种差异的主要原因是,在不同大小的孔洞附近,位错的可动性和位错源的可激活性不同;(3)在微米尺度下,特别是孔洞半径足够小时,孔洞长大的呈现出明显的离散性,这些现象难以被高阶连续理论所捕捉。
5、基于位错发射的Rice-Thomson模型,研究了不同大小、不同取向的椭圆孔洞表面的位错发射及由此导致的孔洞长大机制。通过对单晶体中椭圆孔洞表面位错发射的分析得到:(1)存在一个临界应力,当外载低于该临界应力时,孔洞表面基本不发生位错发射,孔洞难以长大,当外载大于该临界应力时,孔洞表面发射位错并引起孔洞的突发性长大;(2)在纳米量级下,椭圆孔洞位错发射的临界应力呈现出尺度效应,孔洞的等效半径越小,位错发射所需临界应力就越大,而在同一等效半径下,椭圆孔洞的位错发射临界应力较圆孔洞小得多;(3)位错发射及由此引起的孔洞长大与椭圆孔洞主轴的取向有很大关系。
6、基于二维离散位错动力学模拟方法,研究了无限大FCC单晶体内微夹杂周围的应力场分布,讨论了夹杂处孔洞形核的可能机制。通过对等轴拉伸载荷下含微夹杂的FCC单晶体的离散位错模拟得到:(1)由于基体/夹杂界面附近产生的位错障碍和位错塞积,基体/夹杂界面上出现一系列的应力峰;(2)随着夹杂半径的减小,界面上应力峰的数目和应力峰值都不断减小。这些结果表明,微尺度下,孔洞在基体/夹杂界面形核可能与界面上应力峰的数目和峰值的大小有关,并且夹杂越小,孔洞形核越难,这与现有实验结果定性一致。
With the rapid development of Micro Electromechanical System (i.e. MEMS) in recent years, various micro-apparatus and micro-machines have been widely used. A series of micro-mechanical tests and analytic results have already indicated that the mechanical behaviors show strong scale effect when the sample falls within the micron size range. At the micron scale, the strength analyses and structure optimization based on the classical independent-scale constitutive relation usually fail to be efficient. Thus, deep and systemic researches on damage and deformation behavior at the micron scale are of fundamental importance not only in understanding the damage mechanism of solids but also in engineering application.
In the present dissertation, the micro-scale effect on damage and deformation behavior is investigated by both the strain gradient theory and the discrete dislocation dynamic modeling. The results from two kinds of methods are complementary with each other, which introduce more information about the subject to us. The main results are shown as follows:
Based on the SG strain gradient deformation theory and the infinite model with an isolated spheroidal void, the size and shape effects on void growth in triaxial stress field are carefully investigated. It is found that (1) there exists a“critical void radius”, the void is difficult to grow by plastic flow in the matrix material when the equivalent radius of micro-void approach or is lower than the“critical radius”; (2) the“critical radius”is independent of the void shape and the remote stress statue, and approximately equals to the material intrinsic length associated with the stretch strain gradient; (3)the void size effect and shape effect is coupled with each other, generally, the shape effect can enhance the scale effect. In addition, this coupling is closely related to the remote stress triaxiality.
Based on the strain gradient deformation theory and the representative volume element (RVE) model, a scale-dependent macroscopic plastic potential of porous material is deduced, which generalizes application of the classical Gurson model to the micron size range. Further analyses show that(1)at a given void volume fraction, the growth rate of micron sized void is lower than that predicated by the traditional R-T model and Gurson model; (2) the yield locus of porous materials expands outwards with the void size decreasing.
The effects of particle size and shape on the stress distribution at the particle/matrix interface and within the particle are investigated by employing the scale-dependent strain gradient SG theory to provide a better understanding to the micro-particle reinforcement mechanism. By solving a axial-symmetrical boundary value problem consisting of a infinite solid and an isolated spheriodal particle embedded in an infinite matrix, some interesting results are found as follow: (1) at the micron scale, the smaller the particle is, the higher the stresses at the matrix/inclusion interface and within the particle are; (2) this size effect is more remarkable with decreasing the stress triaxiality and increasing the remote equivalent strain. To reasonably equilibrate the double stress traction at the particle-matrix interface, a new parameter named“interfacial energy density”is introduced. The influence of“interfacial energy density”on the stress distribution is also investigated.
A program for discrete dislocation dynamic modeling is developed. Using this program, the growth of an isolated void embedded in an infinite FCC single crystal is investigated. The emphasis is focused on the size effect of void growth and its intrinsic mechanism. The calculation results show that: (1) the expansion of dislocation shear loop and reaching the void surface is main mechanism controlling the void growth in single crystal; (2) the void growth vs. the remote strain curves are continuous and smooth for the larger voids, but there are some steps on the curves for the smaller void, so the internal mechanism controlling the void growth is different for the voids with various sizes; (3)this size effect is closely associated with the dislocation mobility and the number of dislocation sources activated around the void; (4) the discreteness of void growth is inherent and significant for the micron sized void.
Based on the well-known Rice-Thomson model, dislocation emission and void growth induced by it are considered for elliptic voids with various sizes and different directions. The results indicate that: (1) there exist a“critical stress”, when the applied stress is lower than this“critical stress”, dislocations can not be emitted from the void surface, while when the applied stress is larger than the“critical stress”, the dislocation emission is activated and can introduce the nano-void to grow suddenly; (2) within the nanometer size range, the dislocation emission“critical stress”for elliptic void is size dependent, i.e. the smaller the void is, the higher the critical stress is, and at a given equivalent radius, the critical stress for the elliptic voids is lower than that for the circular voids; (3) the dislocation emission and the void growth induced by it are closely related to the direction angle between the slip plane and the axis of the elliptic void.
By employing the 2D discrete dislocation dynamic modeling, the stress distribution in the vicinity of an isolate circular particle embedded in an infinite FCC matrix is investigated, and the void nucleation mechanism is discussed. The results are shown that: (1) due to the dislocation obstruction and pile-ups near the interface, there are a series of peaks for the interfacial normal stress at the particle-matrix interface; (2) the number of interfacial stress peak and the magnitude of these peaks increases with increasing the inclusion radius. These results indicate that, within the micron scale range, the void nucleation at the matrix/particle interface seems to be related to the number and magnitude of the interfacial stress peak. According to this assumption, some experiment observations can be explained, i.e. the smaller the particle is, the more difficult the void nucleation is.
引文
[1] 陈少华;王自强.应变梯度理论进展.力学进展,2003,33(2):207-216.
[2] Lloyd, DJ. Particle reinforced aluminum and Magnesium matrix composites. Int Mater Rev, 1994, 39(1):1-23.
[3] Ma, Q; Clark, DR. Size dependent hardness of silver single crystals. J Mater Res, 1995, 10(4):853-863.
[4] Nix, WD; Gao, H. Indentation size effects in crystalline materials: A law for strain gradient plasticity. J Mech Phys Solids, 1998, 46(3): 411-425.
[5] Wei, Y; Hutchinson, JW. Hardness trends in micron scale indentation, J Mech Phys Solids. 2003, 51(11-12):2037-2056.
[6] Fleck, NA; Muller, GM; Ashby, MF; et al. Strain gradient plasticity: theory and experiment. Acta Mater, 1994, 42(2):475-487.
[7] Stolken, JS; Evans, AG. A Microbend test method for measuring the plasticity length scale. Acta Mater, 1998, 46(14):5109-5115.
[8] Wang,W; Huang, Y; Hsia, KJ; et al. A study of microbend test by strain gradient plasticity. Int J of Plasticity, 2003, 19(3):365-382.
[9] Shu, JY; Barlow, CY. Strain gradient effects on microscopic strain field in a metal matrix composite. Int J of Plasticity, 2000, 16(5):563-591.
[10] Schluter, N; Grimpe, F; Bleck, W; et al. Modeling of the damage in ductile steels. Comp Mater Sci 1996, 7(1-2):27-33
[11] Khraishi, TA; Khaleel, MA. A parametric-experimental study of void growth in superplastic deformation. Int J of Plasticity, 2001, 17(3):297-315.
[12] Douglass, MR. Lifetime estimates and unique failure mechanisms of the digital micromirror device (DMD). In: Annual Proceedings-reliability physics symposium. Piscataway, NJ, USA: IEEE, 1998. 9-16.
[13] Gao, H; Huang, Y; Nix, WD; et al. Mechanism-based strain gradient plasticity--I Theory. J Mech Phys Solids, 1999, 47(6):1239-1263.
[14] Hutchinson, JW. Plasticity at micron scale. Int J of solids & structures, 2000, 37(1-2):225-238.
[15] Arzt, E. Size effects in materials due to microstructural and dimensional constraints: A comparative review. Acta Mater, 1998, 46(16):5611-5626.
[1] 陈少华;王自强.应变梯度理论进展.力学进展,2003,33(2):207-216.
[2] Lloyd, DJ. Particle reinforced aluminum and Magnesium matrix composites. Int Mater Rev, 1994, 39(1):1-23.
[3] Ma, Q; Clark, DR. Size dependent hardness of silver single crystals. J Mater Res, 1995, 10(4):853-863.
[4] Nix, WD; Gao, H. Indentation size effects in crystalline materials: A law for strain gradient plasticity. J Mech Phys Solids, 1998, 46(3): 411-425.
[5] Wei, Y; Hutchinson, JW. Hardness trends in micron scale indentation, J Mech Phys Solids. 2003, 51(11-12):2037-2056.
[6] Fleck, NA; Muller, GM; Ashby, MF; et al. Strain gradient plasticity: theory and experiment. Acta Mater, 1994, 42(2):475-487.
[7] Stolken, JS; Evans, AG. A Microbend test method for measuring the plasticity length scale. Acta Mater, 1998, 46(14):5109-5115.
[8] Wang,W; Huang, Y; Hsia, KJ; et al. A study of microbend test by strain gradient plasticity. Int J of Plasticity, 2003, 19(3):365-382.
[9] Shu, JY; Barlow, CY. Strain gradient effects on microscopic strain field in a metal matrix composite. Int J of Plasticity, 2000, 16(5):563-591.
[10] Schluter, N; Grimpe, F; Bleck, W; et al. Modeling of the damage in ductile steels. Comp Mater Sci 1996, 7(1-2):27-33
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