平面应力准静态扩展裂纹尖端场的弹粘塑性分析
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摘要
裂纹尖端场是断裂力学研究的重要课题之一。在扩展裂纹尖端,一方面由于应变奇异性的存在,会产生较高的应变率;另一方面高度的能量集中导致不可逆变形,大部分变形能以热的形式释放出来,裂纹尖端局部出现高温现象。在这样的高应变率以及高温情况下,固体材料性质发生变化,粘性流动在裂纹尖端的形变中所占的比例相对增加,会同时出现弹性、粘性和塑性性质。因此,在研究扩展裂纹尖端渐近场时,应该考虑到材料的粘性效应,这不仅更加符合实际情况,而且可能因此而解决此前忽略粘性效应所得解中存在的一些问题。
本文考虑粘性效应,采用高玉臣提出的弹粘塑性模型,通过对粘性系数合理的假设,经过渐近分析推导得出材料的一种率敏感型本构关系。采用这种率敏感型本构关系,本文对平面应力I型准静态扩展裂纹的尖端场进行渐近分析。引入Airy应力函数,由应变率变形协调方程得到裂尖场的运动控制方程,根据I型裂纹对称性及表面自由条件给出问题边界条件,通过选取适当的特征参数数值,对控制方程进行数值求解,得到完全连续的裂纹尖端应力场的角分布曲线。具体分析了渐近解的性质,并讨论解随各特征参数的变化规律。
由裂纹尖端场控制方程的推导发现,对于准静态扩展裂纹的平面应力问题,泊松比不出现在控制方程中,这与平面应变问题不同。裂尖场具有幂奇异性,是完全连续的自治场。在靠近裂纹界面处,质点由拉伸状态变为压缩状态。裂尖场应力和塑性应变幅值随材料硬化增强而减小,随粘性增强而增大。材料的粘性是裂尖场的主控参数,不仅主导裂纹尖端应力和应变场的强度,而且对裂尖场的分区构造有明显影响。材料的硬化系数对裂尖场的分区构造有一定影响,但不是很明显。当硬化系数为零时,材料退化为粘弹理想塑性,本文的线性硬化解退化为相应的理想塑性解。
The research of crack-tip field is one of the most important tasks of fracture mechanics. High strain rate will occur at the tip of a growing crack due to the existence of strain singularity. Furthermore, the high energy concentrations at a moving crack-tip will cause irreversible deformation and a great amount of energy of deformation is released in the form of heat which can raise the temperature at the crack-tip as high as a thousand degree. As a consequence, the viscosity of material is an important factor in the study of singular field of the crack-tip. So the viscosity effect should be considered in order to solve these problems better.
The viscosity is considered in the dissertation, with the adoption of elastic viscoplastic model presented by Y. C. Gao to describe the stress-strain relation of the material at the crack-tip. With a rational assumption of the viscosity coefficient of the material, the exponent of singularity is determined through asymptotic analyses, and the rate-sensitive constitutive equation is derived under the model. With the adoption of the rate-sensitive constitutive relationship, it is asymptotically investigated the propagating tip fields of plane stress mode I crack, and the quasi-static equations are obtained governing the stress fields at the crack-tip. Numerical calculations of governing equations are carried out with selections of appropriate values of characteristic parameter by combinations of boundary conditions, and the fully continuous stress-strain fields are obtained at the crack-tip. The nature of asymptotic solution is analyzed and the variations of solutions are discussed according to each parameter.
We can see that poisson’s ratio doesn’t exist in the quasi-static equations of plane stress I model crack governing the stress field, which is different from that of plane strain. The crack-tip field has power singularity, and is continues and self-rule. Near the crack interface, the particles are transferred from compression to tension. The amplitude of crack-tip stress field decreases when the coefficient of hardening becomes bigger, and increases when viscosity increases. Furthermore, viscosity dominates the structure of crack-tip field. The material turns to the ideal-plastic material when the hardening coefficient is zero. The solutions are the same with that of quasi-static HR solutions under the condition of quasi-static propagation.
本文考虑粘性效应,采用高玉臣提出的弹粘塑性模型,通过对粘性系数合理的假设,经过渐近分析推导得出材料的一种率敏感型本构关系。采用这种率敏感型本构关系,本文对平面应力I型准静态扩展裂纹的尖端场进行渐近分析。引入Airy应力函数,由应变率变形协调方程得到裂尖场的运动控制方程,根据I型裂纹对称性及表面自由条件给出问题边界条件,通过选取适当的特征参数数值,对控制方程进行数值求解,得到完全连续的裂纹尖端应力场的角分布曲线。具体分析了渐近解的性质,并讨论解随各特征参数的变化规律。
由裂纹尖端场控制方程的推导发现,对于准静态扩展裂纹的平面应力问题,泊松比不出现在控制方程中,这与平面应变问题不同。裂尖场具有幂奇异性,是完全连续的自治场。在靠近裂纹界面处,质点由拉伸状态变为压缩状态。裂尖场应力和塑性应变幅值随材料硬化增强而减小,随粘性增强而增大。材料的粘性是裂尖场的主控参数,不仅主导裂纹尖端应力和应变场的强度,而且对裂尖场的分区构造有明显影响。材料的硬化系数对裂尖场的分区构造有一定影响,但不是很明显。当硬化系数为零时,材料退化为粘弹理想塑性,本文的线性硬化解退化为相应的理想塑性解。
The research of crack-tip field is one of the most important tasks of fracture mechanics. High strain rate will occur at the tip of a growing crack due to the existence of strain singularity. Furthermore, the high energy concentrations at a moving crack-tip will cause irreversible deformation and a great amount of energy of deformation is released in the form of heat which can raise the temperature at the crack-tip as high as a thousand degree. As a consequence, the viscosity of material is an important factor in the study of singular field of the crack-tip. So the viscosity effect should be considered in order to solve these problems better.
The viscosity is considered in the dissertation, with the adoption of elastic viscoplastic model presented by Y. C. Gao to describe the stress-strain relation of the material at the crack-tip. With a rational assumption of the viscosity coefficient of the material, the exponent of singularity is determined through asymptotic analyses, and the rate-sensitive constitutive equation is derived under the model. With the adoption of the rate-sensitive constitutive relationship, it is asymptotically investigated the propagating tip fields of plane stress mode I crack, and the quasi-static equations are obtained governing the stress fields at the crack-tip. Numerical calculations of governing equations are carried out with selections of appropriate values of characteristic parameter by combinations of boundary conditions, and the fully continuous stress-strain fields are obtained at the crack-tip. The nature of asymptotic solution is analyzed and the variations of solutions are discussed according to each parameter.
We can see that poisson’s ratio doesn’t exist in the quasi-static equations of plane stress I model crack governing the stress field, which is different from that of plane strain. The crack-tip field has power singularity, and is continues and self-rule. Near the crack interface, the particles are transferred from compression to tension. The amplitude of crack-tip stress field decreases when the coefficient of hardening becomes bigger, and increases when viscosity increases. Furthermore, viscosity dominates the structure of crack-tip field. The material turns to the ideal-plastic material when the hardening coefficient is zero. The solutions are the same with that of quasi-static HR solutions under the condition of quasi-static propagation.
引文
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2 J. W. Hutchinson. Singular Behavior at The End of Tensile Crack in A Hardening Material. Journal of Mechanics and Physics of Solids. 1968, 16(1): 13-21
3 J. R. Rice, G. F. Rosengren. Plane Strain Deformation Near Crack Tip in A Power Law Hardening Material. Journal of Mechanics and Physics of Solids. 1968, 16(1): 1-12
4 J. W. Hutchinson. Plastic Stress and Strain Field at A Crack-tip. Journal of Mechanics and Physics of Solids. 1968, 16(4): 337-347
5黄克智,余寿文.弹塑性断裂力学.清华大学出版社,北京, 1985: 90-371
6高玉臣.裂纹起始扩展的弹塑性场.固体力学学报. 1980, 12 (1): 67-76
7靳志和,余寿文.理想弹塑性材料平面应力I型裂纹尖端的弹塑性场.力学学报. 1986, 18 (1): 88-92
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9 A. D. Chitelay, F. A. McClintock. Elastic-plastic Mechanics of Steady Crack Growth under Anti-plane Shear. Journal of Mechanics and Physics of Solids. 1971, 19(3): 147-163
10 L. I. Slepyan. Growing Crack during Plane Deformation of An Elastic-plastic Body. Izv. Akad. Nauk. SSSR. Mekhanika Tverdogo Tela. 1974, 9(1): 57-67
11高玉臣.理想塑性介质中裂纹定常扩展的弹塑性场.力学学报. 1980, 12(1): 48-56
12 J. R. Rice, W. J. Drugan, T. L. Sham. Elastic-plastic Analysis of Growing Crack. Fracture Mechanics. 1980, ASTM STP 700: 189-221
13 J. R. Rice. Elastic-plastic Crack Growth in Mechanics of Solids. 60th Anniversary Volume, Pergamon Press. H. G. Hopkins and M. J. Sewell ed. Oxford, Rodney Hill, 1981: 539-562
14高玉臣.裂纹尖端的弹塑性场.力学学报. 1981年特刊: 100-110
15 Y. C. Gao. The Influence of Compressibility on The Elastic-plastic Field of A Growing Crack. ASTM 2nd International Symposium on Elastic-Plastic Fracture Mechanics. Philadelphia, 1981: 6-10
16 W. J. Drugan, J. R. Rice, T. L. Sham. Asymptotic Analysis of Growing Plane StrainTensile Cracks in Elastic-ideally Plastic Solids. Journal of Mechanics and Physics of Solids. 1982, 30(6): 447-473
17罗学富,黄克智.可压缩弹塑性扩展裂纹尖端场问题的正确提法及其解.中国科学A辑. 1988, 12: 1274-1282
18 P. P. Cas ta~neda. Asymptotic Fields of A Perfectly-plastic Plane Stress Mode II Growing Crack. Journal of Mechanics and Physics of Solids. 1986, 34(6): 831- 833
19 Y. C. Gao. Pseudo Plane Stress Plastic Field for Mode I Crack Growth. Theoretical and Applied Fracture Mechanics. 1997, 27(3): 203-212
20 J. C. Amazigo, J. W. Hutchinson. Crack-tip Fields in Steady Crack Growth with Linear Strain-hardening. Journal of Mechanics and Physics of Solids. 1977, 25(2): 81-97
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23 P. P. Cas ta~neda. Asymptotic Fields in Steady Crack Growth with Linear Strain- Hardening. Journal of Mechanics and Physics of Solids. 1987, 35(2): 227-268
24 W. Yang. Non-uniform Singularity Field for Mode III Crack Propagation in A Slightly Linear Strain Hardening Material. C. Ouyang ed. Proceedings of International Conference on Fracture and Fracture Mechanics. Fudan University Press. Shanghai, 1987: 270-275
25高玉臣,张晓堤,黄克智.幂硬化材料Ⅲ型稳恒扩展裂纹的奇异场.力学学报. 1981, 13(5): 452-463
26 Y. C. Gao, X. T. Zhang, K. C. Hwang. The Asymptotic Near-tip Solution for Mode III Crack in Steady Growth in Power Hardening Media. International Journal of Fracture. 1983, 21(4): 301-317
27 Y. C. Gao. Exponential Stress and Strain Singularity at Tip of Mode I Growing Crack in Power Hardening Material. Theoretical and Applied Fracture Mechanics. 1996, 25(2): 103-111
28 Y. C. Gao. Near-tip Fields for Pseudo Mode II Crack Growth in Power Law Hardening Material. Theoretical and Applied Fracture Mechanics. 1996, 25(3): 179-186
29 Y. C. Gao. Pseudo Plane Stress Mode II Crack Growth in Plastic Materials. Theoreticaland Applied Fracture Mechanics. 1997, 27(3): 193-202
30 C. Y. Hui, H.Riedel. The Asymptotic Stress and Strain Field near The Tip of A Growing Crack under Creep Conditions. International Journal of Fracture. 1981, 17(4): 409-425
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63贾斌,王振清,李永东,梁文彦.线性硬化材料中稳恒扩展裂纹尖端场的粘塑性解.应用数学和力学, 2006, 27(4): 470-476
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2 J. W. Hutchinson. Singular Behavior at The End of Tensile Crack in A Hardening Material. Journal of Mechanics and Physics of Solids. 1968, 16(1): 13-21
3 J. R. Rice, G. F. Rosengren. Plane Strain Deformation Near Crack Tip in A Power Law Hardening Material. Journal of Mechanics and Physics of Solids. 1968, 16(1): 1-12
4 J. W. Hutchinson. Plastic Stress and Strain Field at A Crack-tip. Journal of Mechanics and Physics of Solids. 1968, 16(4): 337-347
5黄克智,余寿文.弹塑性断裂力学.清华大学出版社,北京, 1985: 90-371
6高玉臣.裂纹起始扩展的弹塑性场.固体力学学报. 1980, 12 (1): 67-76
7靳志和,余寿文.理想弹塑性材料平面应力I型裂纹尖端的弹塑性场.力学学报. 1986, 18 (1): 88-92
8章梓茂,马兴瑞,高玉臣.理想塑性介质中平面应力静止裂纹的尖端弹塑性场.力学学报. 1990, 22(2): 229-235
9 A. D. Chitelay, F. A. McClintock. Elastic-plastic Mechanics of Steady Crack Growth under Anti-plane Shear. Journal of Mechanics and Physics of Solids. 1971, 19(3): 147-163
10 L. I. Slepyan. Growing Crack during Plane Deformation of An Elastic-plastic Body. Izv. Akad. Nauk. SSSR. Mekhanika Tverdogo Tela. 1974, 9(1): 57-67
11高玉臣.理想塑性介质中裂纹定常扩展的弹塑性场.力学学报. 1980, 12(1): 48-56
12 J. R. Rice, W. J. Drugan, T. L. Sham. Elastic-plastic Analysis of Growing Crack. Fracture Mechanics. 1980, ASTM STP 700: 189-221
13 J. R. Rice. Elastic-plastic Crack Growth in Mechanics of Solids. 60th Anniversary Volume, Pergamon Press. H. G. Hopkins and M. J. Sewell ed. Oxford, Rodney Hill, 1981: 539-562
14高玉臣.裂纹尖端的弹塑性场.力学学报. 1981年特刊: 100-110
15 Y. C. Gao. The Influence of Compressibility on The Elastic-plastic Field of A Growing Crack. ASTM 2nd International Symposium on Elastic-Plastic Fracture Mechanics. Philadelphia, 1981: 6-10
16 W. J. Drugan, J. R. Rice, T. L. Sham. Asymptotic Analysis of Growing Plane StrainTensile Cracks in Elastic-ideally Plastic Solids. Journal of Mechanics and Physics of Solids. 1982, 30(6): 447-473
17罗学富,黄克智.可压缩弹塑性扩展裂纹尖端场问题的正确提法及其解.中国科学A辑. 1988, 12: 1274-1282
18 P. P. Cas ta~neda. Asymptotic Fields of A Perfectly-plastic Plane Stress Mode II Growing Crack. Journal of Mechanics and Physics of Solids. 1986, 34(6): 831- 833
19 Y. C. Gao. Pseudo Plane Stress Plastic Field for Mode I Crack Growth. Theoretical and Applied Fracture Mechanics. 1997, 27(3): 203-212
20 J. C. Amazigo, J. W. Hutchinson. Crack-tip Fields in Steady Crack Growth with Linear Strain-hardening. Journal of Mechanics and Physics of Solids. 1977, 25(2): 81-97
21 J. W. Hutchinson. Crack-tip Singularity Fields in Nonlinear Fracture Mechanics. A Survey of Current Status. ICF5. Advances in Fracture Research. D. Francois ed. V6, Pergamon Press, 1981: 1847-1872
22 R. F. Zhang, X. T. Zhang, K. C. Hwang. Near-tip Fields for Plane Strain Mode I Steady Crack Growth in Linear Hardening Material with Bauschinger Effect.北京国际断裂力学学会讨论会论文集.科学出版社.北京, 1983: 283-290
23 P. P. Cas ta~neda. Asymptotic Fields in Steady Crack Growth with Linear Strain- Hardening. Journal of Mechanics and Physics of Solids. 1987, 35(2): 227-268
24 W. Yang. Non-uniform Singularity Field for Mode III Crack Propagation in A Slightly Linear Strain Hardening Material. C. Ouyang ed. Proceedings of International Conference on Fracture and Fracture Mechanics. Fudan University Press. Shanghai, 1987: 270-275
25高玉臣,张晓堤,黄克智.幂硬化材料Ⅲ型稳恒扩展裂纹的奇异场.力学学报. 1981, 13(5): 452-463
26 Y. C. Gao, X. T. Zhang, K. C. Hwang. The Asymptotic Near-tip Solution for Mode III Crack in Steady Growth in Power Hardening Media. International Journal of Fracture. 1983, 21(4): 301-317
27 Y. C. Gao. Exponential Stress and Strain Singularity at Tip of Mode I Growing Crack in Power Hardening Material. Theoretical and Applied Fracture Mechanics. 1996, 25(2): 103-111
28 Y. C. Gao. Near-tip Fields for Pseudo Mode II Crack Growth in Power Law Hardening Material. Theoretical and Applied Fracture Mechanics. 1996, 25(3): 179-186
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30 C. Y. Hui, H.Riedel. The Asymptotic Stress and Strain Field near The Tip of A Growing Crack under Creep Conditions. International Journal of Fracture. 1981, 17(4): 409-425
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