岩土材料中球形孔洞膨胀问题的力学分析
|
摘要
岩土材料是自然界中应用最广泛的材料之一,属于典型的压力敏感性材料,由于材料中存在微结构(孔洞、微缺陷、微裂纹等),材料变形和破坏机理复杂,其力学性能与金属材料相比有显著的不同,这主要表现在应力应变关系具有非线性和明显的硬化或软化特征以及静水压力影响材料的屈服、损伤和破坏。由于微、细观结构的相似性,岩土材料本构方程的研究对水泥、混凝土和钢筋混凝土以及泡沫金属等新型材料的研究有着重要的参考价值。
由于球形孔洞膨胀模型具有对称性、简便明确并易于给出应力和应变场解,从而揭示材料的变形本质,因而广泛应用于固体力学、材料科学、固体物理、爆炸力学等学科领域。本文对岩土力学中球形孔洞膨胀问题进行了研究,采用四区模型,分别讨论了静态和动态膨胀条件下塑性区、损伤区、弹性区的应力和位移场的变化情况。由于采用分区思想,在不同区域内,材料的变形机理不同因而采用的本构模型不同。本文的主要工作如下：
1.以球形孔洞膨胀四区模型(塑性区、损伤区、弹性区、应力自由区)为研究对象,认为在塑性区和弹性区之间存在一个损伤区,由于它连接塑性区和弹性区的约束,因此不可能达到完全的破坏,即建立了σθ≠0的四区球形孔洞膨胀模型。
2.根据岩土材料的变形连续性特征,选择椭圆形屈服准则,从宏观塑性力学原理出发,建立了反映岩土材料压力敏感性特征的增量型和全量型本构方程；从细观塑性力学原理出发,给出了微结构对岩土类材料塑性屈服影响的塑性屈服条件；引入Lemaitre有效应力的概念建立了材料的损伤本构方程。
3.采用分区思想,对岩土材料中球形孔洞膨胀静态问题进行了弹塑性和损伤力学分析,给出边界条件和交界处的连续性条件,并分别讨论了塑性区、损伤区、弹性区的应力和位移场的变化与材料参数的关系,为更深刻地认识材料的变形本质提供了参考。
4.采用自相似假设,结合三区模型(损伤区、弹性区、应力自由区)对岩土材料中球形孔洞动态扩展的问题进行弹性-损伤力学分析。用单参数打靶法数值求解了损伤区和弹性区的场量变化,讨论了动态扩展条件下损伤区的影响因素,并指出在一定条件下材料可能被破坏,不能再用损伤本构来描述,此时采用四区模型进行研究将更为合理。
5.采用椭圆型压力敏感性材料屈服准则和自相似假设,在对损伤区研究的基础上,用塑性区继续研究在损伤区域外的场量数值解。通过推导动态扩展条件下增量型(理想塑性材料)和全量型(幂硬化材料)塑性区的非线性微分控制方程组,利用三区模型确定的损伤区边界条件,通过双参数打靶法进行数值求解,并分别讨论了材料参数对场量的影响。
本文综合讨论了球形孔洞膨胀四区模型描述的岩土材料爆炸应力场,对塑性区、损伤区、弹性区的应力和位移场的研究会使人们更深刻地认识岩土材料的变形本质,为理论研究和工程应用奠定基础。
Geomaterials, which belong to typical pressure sensitive materials, are the most widely used materials in the nature. As they contain micro-structures (such as micro-voids, defects and cracks), the deformation and failure mechanism are complicated. Their mechanical properties are significantly different from those of metal materials, which mainly being shown in non-linear stress-strain relations, apparent hardening or softening characteristics, and hydrostatic pressure's effectiveness to their yield, injury and failure. Because having the similarity of micro-meso structure, study on the constitutive equations of geomaterials has important reference value for study of cement, concrete, reinforced concrete, and some other new materials such as foam metals.
As the spherical cavity expansion model has the property of being symmetry, simple, and clear, it can provide the solution of stress and strain field easily so as to reveal the nature of materials'deformation, therefore it is widely used in fields of solid mechanics, materials science, solid state physics and explosion mechanics, etc. In this paper, problems of spherical cavity expansion in geomaterials had been studied by using a four-region model, changes of stress and displacement in plastic, damage and elastic regions were discussed under conditions of static and dynamic expansion. Because using the idea of dividing the model into different regions with different distortion mechanism, different constitutive equations were adopted. The main work of this paper is as follows:
1. Take a four regions' spherical cavity expansion model (plastic region/damage region/elastic region/stress-free region) as a research object, a damage region is thought being existed between plastic region and elastic region. As damage region connects constraints of both plastic region and elastic region, which means it is impossible to achieve complete destruction, we established a four-region spherical cavity expansion model withσθ≠0.
2. Base on the theory of macroscopic plasticity and the continuous character of geomaterials'deformation, elliptical yield criterion are adopted, a increment type constitutive and a full-bore type equation are constructed to reflect materials'pressure sensitivity. And base on the theory of microscopic plasticity, a plastic yield criterion which reflects the effect of micro-structures to geomaterials' yield is deduced. A damage constitutive equation with Lemaitre effective stress is established also.
3. Using the dividing idea mentioned above, plasticity and damage mechanics are adopted to analyze problems of static spherical cavity expansion in geomaterials, boundary conditions and continuity conditions at the junctions are given, changes of stress and displacement field and relationships between these field quantities and the material parameters are discussed in all the regions of plastic, damage and elastic. These provide a reference to gain a deeper understanding of the deformation nature of geomaterials.
4. Via self-similar assumptions, elastic and damage mechanics are adopted to analyze problems of dynamic spherical cavity expansion in geomaterials by combining the three-region (damage zone, elastic zone and stress-free zone) model. Quantity distributions in damage zone and elastic zone are solved by shooting method with a single parameter, and impact factors of damage zone are discussed, from which it is pointed out that the material might be damaged under certain conditions, when a four-region model should be more reasonable to study the problem.
5. Using elliptic yield criterion of pressure sensitive materials and the self-similar assumption, the plastic zone out of damage zone are adopted to analyze the field solution under conditions of the study in damage zone. Nonlinear differential equations of plastic region of the dynamic expansion are derived for both increment type (elastic-perfectly plastic materials) and full-bore type constitutive equation (power-hardening materials). Via three-zone model we discussed before to determine the plastic boundary conditions, numerical simulation are obtained by shooting method with two parameters, and the impact of material parameters are also discussed.
Overall, the explosive fields of geomaterials are studied comprehensively by using four-region model of the spherical cavity expansion, researches on stress and displacement fields in plastic, damage and elastic zone will provide a deeper understanding of the geomaterials' deformation, meanwhile lay a deep foundation for theoretical research and subsequent engineering applications.
由于球形孔洞膨胀模型具有对称性、简便明确并易于给出应力和应变场解,从而揭示材料的变形本质,因而广泛应用于固体力学、材料科学、固体物理、爆炸力学等学科领域。本文对岩土力学中球形孔洞膨胀问题进行了研究,采用四区模型,分别讨论了静态和动态膨胀条件下塑性区、损伤区、弹性区的应力和位移场的变化情况。由于采用分区思想,在不同区域内,材料的变形机理不同因而采用的本构模型不同。本文的主要工作如下：
1.以球形孔洞膨胀四区模型(塑性区、损伤区、弹性区、应力自由区)为研究对象,认为在塑性区和弹性区之间存在一个损伤区,由于它连接塑性区和弹性区的约束,因此不可能达到完全的破坏,即建立了σθ≠0的四区球形孔洞膨胀模型。
2.根据岩土材料的变形连续性特征,选择椭圆形屈服准则,从宏观塑性力学原理出发,建立了反映岩土材料压力敏感性特征的增量型和全量型本构方程；从细观塑性力学原理出发,给出了微结构对岩土类材料塑性屈服影响的塑性屈服条件；引入Lemaitre有效应力的概念建立了材料的损伤本构方程。
3.采用分区思想,对岩土材料中球形孔洞膨胀静态问题进行了弹塑性和损伤力学分析,给出边界条件和交界处的连续性条件,并分别讨论了塑性区、损伤区、弹性区的应力和位移场的变化与材料参数的关系,为更深刻地认识材料的变形本质提供了参考。
4.采用自相似假设,结合三区模型(损伤区、弹性区、应力自由区)对岩土材料中球形孔洞动态扩展的问题进行弹性-损伤力学分析。用单参数打靶法数值求解了损伤区和弹性区的场量变化,讨论了动态扩展条件下损伤区的影响因素,并指出在一定条件下材料可能被破坏,不能再用损伤本构来描述,此时采用四区模型进行研究将更为合理。
5.采用椭圆型压力敏感性材料屈服准则和自相似假设,在对损伤区研究的基础上,用塑性区继续研究在损伤区域外的场量数值解。通过推导动态扩展条件下增量型(理想塑性材料)和全量型(幂硬化材料)塑性区的非线性微分控制方程组,利用三区模型确定的损伤区边界条件,通过双参数打靶法进行数值求解,并分别讨论了材料参数对场量的影响。
本文综合讨论了球形孔洞膨胀四区模型描述的岩土材料爆炸应力场,对塑性区、损伤区、弹性区的应力和位移场的研究会使人们更深刻地认识岩土材料的变形本质,为理论研究和工程应用奠定基础。
Geomaterials, which belong to typical pressure sensitive materials, are the most widely used materials in the nature. As they contain micro-structures (such as micro-voids, defects and cracks), the deformation and failure mechanism are complicated. Their mechanical properties are significantly different from those of metal materials, which mainly being shown in non-linear stress-strain relations, apparent hardening or softening characteristics, and hydrostatic pressure's effectiveness to their yield, injury and failure. Because having the similarity of micro-meso structure, study on the constitutive equations of geomaterials has important reference value for study of cement, concrete, reinforced concrete, and some other new materials such as foam metals.
As the spherical cavity expansion model has the property of being symmetry, simple, and clear, it can provide the solution of stress and strain field easily so as to reveal the nature of materials'deformation, therefore it is widely used in fields of solid mechanics, materials science, solid state physics and explosion mechanics, etc. In this paper, problems of spherical cavity expansion in geomaterials had been studied by using a four-region model, changes of stress and displacement in plastic, damage and elastic regions were discussed under conditions of static and dynamic expansion. Because using the idea of dividing the model into different regions with different distortion mechanism, different constitutive equations were adopted. The main work of this paper is as follows:
1. Take a four regions' spherical cavity expansion model (plastic region/damage region/elastic region/stress-free region) as a research object, a damage region is thought being existed between plastic region and elastic region. As damage region connects constraints of both plastic region and elastic region, which means it is impossible to achieve complete destruction, we established a four-region spherical cavity expansion model withσθ≠0.
2. Base on the theory of macroscopic plasticity and the continuous character of geomaterials'deformation, elliptical yield criterion are adopted, a increment type constitutive and a full-bore type equation are constructed to reflect materials'pressure sensitivity. And base on the theory of microscopic plasticity, a plastic yield criterion which reflects the effect of micro-structures to geomaterials' yield is deduced. A damage constitutive equation with Lemaitre effective stress is established also.
3. Using the dividing idea mentioned above, plasticity and damage mechanics are adopted to analyze problems of static spherical cavity expansion in geomaterials, boundary conditions and continuity conditions at the junctions are given, changes of stress and displacement field and relationships between these field quantities and the material parameters are discussed in all the regions of plastic, damage and elastic. These provide a reference to gain a deeper understanding of the deformation nature of geomaterials.
4. Via self-similar assumptions, elastic and damage mechanics are adopted to analyze problems of dynamic spherical cavity expansion in geomaterials by combining the three-region (damage zone, elastic zone and stress-free zone) model. Quantity distributions in damage zone and elastic zone are solved by shooting method with a single parameter, and impact factors of damage zone are discussed, from which it is pointed out that the material might be damaged under certain conditions, when a four-region model should be more reasonable to study the problem.
5. Using elliptic yield criterion of pressure sensitive materials and the self-similar assumption, the plastic zone out of damage zone are adopted to analyze the field solution under conditions of the study in damage zone. Nonlinear differential equations of plastic region of the dynamic expansion are derived for both increment type (elastic-perfectly plastic materials) and full-bore type constitutive equation (power-hardening materials). Via three-zone model we discussed before to determine the plastic boundary conditions, numerical simulation are obtained by shooting method with two parameters, and the impact of material parameters are also discussed.
Overall, the explosive fields of geomaterials are studied comprehensively by using four-region model of the spherical cavity expansion, researches on stress and displacement fields in plastic, damage and elastic zone will provide a deeper understanding of the geomaterials' deformation, meanwhile lay a deep foundation for theoretical research and subsequent engineering applications.
引文
[1]陈顒,黄庭芳.岩石物理学.北京：北京大学出版社,2001：89—102页
[2]谢和平,陈忠辉.岩石力学.北京：科学出版社,2004：46—64页
[3]李翼祺,马素贞著.爆炸力学.北京：科学出版社,1992：356—417页.
[4]杨桂通.土动力学.北京：中国建材工业出版社,2000：92—134页.
[5]张晓春等.岩体裂纹演化及其力学特性的研究进展.力学进展.1999,29(1)：97—104页
[6]陈惠发.土木工程材料的本构关系—塑性与建模.武汉：华中科技大学出版社,2001：212—267页
[7]Gibson L J, Ashby M F. Cellular Solids:Structure and Properties[M].Oxford:Pergamon Press,1997:32-50P
[8]Gibson L J. Mechanic behavior of metallic foam. Annual Review of Materials Science,2000,30:191-227P
[9]Bishop R F, Hill R, Mott N F. The theory of indentation and hardness. Test. Proc. Phys. Soc.1945,57(3):147-159P.
[10]Taylor G I. The formation and enlargement of a circular hole in a thin plastic sheet. Quarterly. J. Mech. and Appl. Math.,1948,1:103-124P
[11]Hill R. The Mathematical Theory of Plasticity. London. Oxford University Press,1950: 89-101P
[12]Hopkins H G. Dynamic expansion of spherical cavities in metal. Progress in Solid Mechanics, Edited by Sneddon I N, Hill R. North-Holland, Amsterdam,1960.
[13]Macek R W, Duffey T A. Finite cavity expansion method for near-surface effects and layering during Earth penetration. International Journal of Impact Engineering,2000, 24(3):239-258P
[14]Rosenberg Z, Dekel E. A numerical study of the cavity expansion process and its application to long-rod penetration mechanics. International Journal of Impact Engineering. No.35,2008:147-154P
[15]俞茂宏.强度理论百年总结.力学进展.2004,34(4)：529—560页
[16]Jianchun Li, Guowei Ma, Maohong Yu. Penetration analysis for geo-material based on unified strength criterion. International Journal of Impact Engineering.2008,35:1154 -1163P
[17]Maohong Yu, Guiyun Xia, V A Kolupaev. Basic characteristics and development of yield criteria for geomaterials. Journal of Rock Mechanics and Geotechnical Engineering,2009,1(1):71-88P
[18]余寿文,冯西桥.损伤力学.北京：清华大学出版社,1997：78—114页
[19]Mura T. Miero mechanics of Defects in Solids, Martinus Nijhoff Publishers,1987: 53-61P
[20]王自强,段祝平.塑性细观力学.北京：科学出版社,1995：242—275页
[21]杨卫.宏细观断裂力学.北京：国防工业出版社,1995：134—288页
[22]卡恰诺夫,连续介质损伤力学.哈尔滨：哈尔滨工业大学出社,1989：1—10P,123—125页
[23]Rabotnov Y N. Creep problems in structural members. North-Holland, Amsterdam,1969.
[24]Lemaitre J, Chaboche J L. Mechanics of Solid Materials, Cambridge:Cambridge University Press,1988:317-329P
[25]Lemaitre J. A course on damage mechanics. Springer-Verlag, Belin.1992.
[26]Hult J and Broberg H. Creep rupture under cyclic lading. Proc. of the second Bulgarran Congress on Mechanics,1976:263-272P
[27]Leckie F A and Hayhurst D R. Constitutive equation of creep rupture. Acta Meta, 1977(25):1059-1076P
[28]Murakami S. Progress of damage mechanics. JSME Inter. Journal.1987,30, No.263: 701-710P
[29]Gurson A L. Continuum theory of ductile rupture by the void nucleation and growth. J. Engng. Mater. Technology,1977,99:234-245P
[30]McClintock F A. A criterion for ductile fracture by the growth of holes. J. Appl. Mech.1968,35:363-371P
[31]Rice I R and Tracey D M. On the ductile enlargement of voids in triaxil stress fields. J. Mech. Phys. Solids,1969,17:201-227P.
[32]Lee J H, Chung J. Yield functions and flow rules for porous pressure-dependent strain-hardening polymeric materials, ASME, Journal of Applied Mechanics,2000, 67:288-297P
[33]Xia L and Shih C F. Ductile crack growth-I, A numerical study using computational cells with micro-structurally based length scales. J. Mech. Phys. Solids.,1995(43): 233-259P
[34]Xia L, Shih C F. Ductile crack growth-Ⅱ, A numerical study using computational cells with micro-structurally based length scales. J. Mech Phys Solids,1995(43):1953-1981P
[35]Tvergaard V, Needleman A. Effects of non-local damage in porous plastic solids. International Journal of Solids and Structures,1995(32):1063-1077P
[36]Needleman A, Tvergaard V. Micromechanical analysis of the ductile-brittle transition. Engineering Fracture Mechanics,1999(62):317-338P
[37]Biglari F R, Nikbin K M, O'Dowd N P. Numerical simulation of crack growth based on void volume fraction. Proceedings of the 10th annual mechanical engineering conference, ISME2002, Tehran Iran,2002,5:492-496P
[38]Tvergaard V, Hutchinson J W. Two mechanisms of ductile fracture:void by void growth versus multiple void interactions. Int. J. Solids Structures,2002,39:3581-3597P.
[39]Tvergaard V, Niordson C. Nonlocal plasticity effects on interaction of different size voids. International Journal of Plasticity,2004,20(1):107-120P
[40]Tvergaard V, Needleman A. Three-dimensional micro-structural effects on plane strain ductile crack growth. Int J Solids Struct,2006,43(20):6165-6179P
[41]董杰,李永池,陈学东.一种新型微孔洞损伤模型.高压物理学报.2007,21,(4)：414—418页
[42]董杰,李永池,陈学东.微孔洞唯象损伤力学模型及其应用.爆炸与冲击.2008,8(5)：443-447页
[43]王自强,秦嘉亮.含空洞非线性材料的本构势和空洞扩展率.固体力学学报,1989,10(2)：127—141页
[44]黄筑平,孙立志.韧性材料中的一个新的动态孔洞增长模型.中国科学A辑,1992(9)：960—969页
[45]Ashby M F, Hallam S D. The failure of brittle solids containing small cracks under compressive stress states. Acta Metall.1986,34:497-510P.
[46]Horii H S, Nemat-Nasser. Overall moduli of solids with microcracks:load-induced anisotropy. J. Mech. Phys.Solids.1983,31:155-171P.
[47]Horii H S, Nemat-Nasser. Brittle fracture in compression:splitting, faulting and brittle-ductile transition. Phil. Trans. Roy. Soc, London:1986, A:337-374P.
[48]Budiansky B, O'Connell R J. Elastic module of a cracked solids. Int. J. Solids Structures,1976,12:81-97P.
[49]高玉臣,朱葳.含微裂纹材料的损伤理论.力学学报,1987,19：541—54页.
[50]唐立强,陈会军,姜民政.海冰板的环形屈曲和压缩破坏.海洋工程,2001,19,(4)：29—33页
[51]赵吉坤,张子明,仝兴华,薛世峰.岩石宏细观弹塑性损伤破坏对比研究.中国石油大学学报(自然科学版),中国石油大学学报(自然科学版)2007,31：78—84页
[52]黄筑平,杨黎明,潘客麟.材料的动态损伤和失效.力学进展,1993,23(4)：433—467页
[53]Butterfield R and Banerjee P K. The effects of pore water pressures on the ultimate bearing capacity of driven piles. In:Proc 2nd south East Asian regional Soil Mech Found Engrg, Tykyo,1970,385-394P
[54]Vesic A S. Expansion of Cavity in an Infinite Soil Mass, J.S.M. F.E. ASCE,1972, 98(3):265-289P
[55]Carter J P, Booker J R and Yeung S K. Cavity Expansion in Cohesive Frictional Soil. Geotechnique 1986,36(3):349-353P
[56]Randoph M F, Carter L P, Wroth C P. Driven Piles in Clay-the Effects of Installation and Subsequent Consolidation. Geotechnique,1979,29(4):361-393P
[57]Salgado R, Mitchel J K, Jamiolkowski. Cavity expansion and penetration resistance in sand. Jour Geotech Geoenvior Engng, ASCE,1997,123(4):344-357P
[58]Chow Y K and Thec I. A Theoretical Study of Pile Heave. Geotechnique,1990,40(1): 1-14P
[59]Mantaras F M, Schnaid F. Cylindrical cavity expansion in dilatants cohesive—frictional materials. Geotechnique,2002,52(5):337-348P
[60]Yu H S, Houlsby G T. Finite cavity expansion in dilatant soils:loading analysis. Geotechnique,1991,41(2):173-183P
[61]Cao L F, Teh C I, Chang M F. Undrained cavity expansion in modified Cam clay Ⅰ: theoretical analysis[J]. Geotechnique,2001,51(4):323-334P
[62]Chang M F, Teh C I, Cao L F. Undrained cavity expansion in modified Cam-clay Ⅱ: application to the interpretation of the piezocone test. Geotechnique,2001,51(4): 335-350P
[63]李月健,陈云敏等,土体内空穴球形扩张问题的一般解及应用.土木工程学报,2002, 35(1)：93-98页
[64]叶建忠,周健.砂土中桩的刺入变形与端阻力发挥的初步研究.建筑结构.2007,Vol37,No.8：P88—90页
[65]刘俊伟,张明义,赵洪福,王静静.基于球孔扩张理论和侧阻力退化效应压桩力计算模拟.岩土力学,Vol.30,No.4,2009：1181—1185页
[66]李赞成,胡功笠,梁济丰,张韬.考虑动态参数的混凝土侵彻工程计算.力学与实践,Vol.31,No.6,2009：42—45页
[67]罗战友,夏建中,龚晓南.不同拉压模量及软化特性材料的柱形孔扩张问题的统一解.工程力学,2008,25(9)：79—84页
[68]汪鹏程,张红亚.弹塑脆性岩土材料中球形孔大应变扩张分析.安徽建筑工业学院学报(自然科学版),2008,16(2)：18—22页
[69]Zhou Aizhao, Lu Tinghao. Elasto—plastic constitutive model of soil—structure interface in consideration of strain softening and dilation. Acta Mechanics Solid Sinica, 2009,22(2):171-179P
[70]胡士兵,王金昌,朱向荣.线性软化土体中球孔扩张问题的解析解.浙江大学学报(工学版),2007,41(9)：1503—1507页
[71]林政,陈云敏,陈仁朋.球形空腔固结解析解及其在地基原位固结系数测试中的应用.岩石力学与工程学报,2005,24(2)：5863—5867页
[72]Feiberger W. A problem in dynamic plasticity:the enlargement of a circular of ho 1 e in a flat sheet. Proc. of the Cambridge Philosophical Society,1952, V01(48):135 P
[73]Forrestal M J, and Luk V K. Dynamic Spherical Cavity—Expansion in a Compressible Elastic-Plastic Solid, J. Appl. Mech.,1988,55:275-279P
[74]Luk V K, Amos D E. Dynamic cylindrical cavity expansion of compressible strain-hardening materials. J. Appl. Mech.,1991,58(2):334-340P
[75]Forrestal M J, Brar N S, LukVK. Penetration of strain— hardening targets with rigid spherical-nose rods. J. Appl. Mech.,1991,58:7-10P
[76]Forrestal M J, Tzou D Y, Askari E, Longcope D B. Penetration into ductile metal targets with rigid spherical—nose rods. Int. J. Impact Eng.,1995,16(5-6):699-710P
[77]Forrestal M J and Tzou D Y. A spherical cavity expansion penetration model for concrete targets. International Journal of Plasticity,1998,14(1-3):173-191P
[78]Satapathy S. Dynamic spherical cavity expansion in brittle ceram -ics. International Journal of Solids and Structures,2001(38):5833-5845P
[79]Satapathy S and Blss S J. Cavity expansion resistance of in brittle materials obeying a two curve pressure-shear behavior. J. Appl. Phys.2000,88(7):4004-4012P
[80]Rosenberg Z, Dekel E. A numerical study of the cavity expansion process and its application to long-rod penetration. International Journal of Impact Engineering,2008, 35:147-154P
[81]Masri R, Durban D. Dynamic spherical cavity expansion in an elastoplastic compressible mises solid. J. Appl. Mech.,2005,72(6):887-898P
[82]Masri R, Durban D. Dynamic cylindrical cavity expansion in an incompressible elastoplastic medium. Acta Mech,2006,181(1-2):105-123P
[83]Masri R, Durban D. Cylindrical cavity expansion in compressible mises and tresca solids. European Journal of Mechanics A/Solids 2007,26(4):712-727P
[84]Narasimhan R. Analysis of indentation of pressure sensitive plastic solids using the expanding cavity model. Mech. Mater.2004,36:633-645P
[85]何涛,文鹤鸣.球形弹对金属靶板侵彻问题的数值模拟.爆炸与冲击,Vol.26,No.5,2006,9：456—461页
[86]蒋志刚,曾首义,周建平.中等厚度金属靶板的三阶段贯穿模型.兵工学报,2007,Vol28,No9：1046—1052页
[87]吴昊,方秦,龚自明,相恒波.应用改进的双剪强度理论分析岩石靶体的弹体侵彻深度.工程力学,2009,Vol.26,No.8：216—222页
[88]Frew D J, Forrestal M J, Cargile J D. The effect of concrete target diameter on projectile deceleration and penetration depth. International Journal of Impact Engineering,2006,32:1584-1594P
[89]韩丽,高世桥,李明辉,刘海鹏.弹丸垂直贯穿混凝土靶的数值研究.北京理工大学学报,2006,26(11)：953—956页
[90]Kogat I, Eslion I. Elastic-plastic contact analysis of a sphere and a rigid flat. J. Appl. Mech.,2002,69(5):657-662 P
[91]Fischer-Cripps A C. Elastic-plastic behavior in materials loaded with a spherical indenter. J. Mater. Sci.,1997,32(3):727-736 P
[92]Mata M, Alcala J. Mechanical property evaluation through sharp indentation in elastoplastic and fully plastic contact regimes. J. Mater. Res.,2003,18(9): 1705-1709P
[93]Mata M, Anglada M and Alcala J. A hardness equation for sharp indentation of elastic-power-law strain-hardening materials. Philos. Mag.2002,82(10):1831-1839P
[94]Mesarovic S D, Fleck N A. Spherical indentation of elastic-plastic solids. Proc. R. Soc. Lond.,1999,455:2707-2728P
[95]Park Y J, Pharr G M. Nanoindentation with spherical indenters:finite element studies of deformation in the elastic-plastic transition regime. Thin Solid Films,2004, 447-448(30):246-250P
[96]Wei Y and Hutchinson J W. Hardness trends in micron scale indentation. J. Mech. Phys. Solids,2003,51,2037-2056P.
[97]Johnson K L. Contact Mechanics. Cambridge:Cambridge University Press,1985: 175P
[98]Chadwick P. The quasi-static expansion of a spherical cavity in metals and ideal soils. Quart. J. Mech. Appl. Math.,1959,12(1):52-71P
[99]Durban D, Baruch M. On the problem of a spherical cavity in an infinite elasto-plastic medium. J. Appl. Mech.,1976,43:633-638P
[100]Durban D, Kubi M. A general solution for the pressurized elastoplastic tube. J. Appl. Mech.,1992,59:20-26P
[101]Durban D, Papanastasiou P. Cylindrical cavity expansion and contraction in pressure sensitive geomaterials. Acta Mech.,1997,122(1-4):99-122P
[102]Brignoni D, Ludiero F. The quasi-static finite cavity expansion in a non-standard elasto-plastic medium. Int. J. Mech. Sic.,1989,31(11-12):825-837P
[103]Jin Ming. Huang kefu. Jike wu. A study of the cafastrophe and the caritation for a spherical cavity in Hook's material with 1/2 poision'ratio. Applied Meth Mech,1999 20(8):928-935P
[104]程昌钧,聂波.幂硬化材料和超弹性材料含球体中空穴的动态生成.固体力学学报,2005,vol26(3)：273—279页
[105]任九生,程昌钧.受内压热超弹性球壳的不稳定性.上海大学学报(自然科学版),2007,13(6)：732—740页
[106]Wei X Y, Zhao Z Y, Gu J. Numerical simulations of rock mass damage induced by underground explosion. International Journal of Rock Mechanics & Mining Sciences, 2009,46:1206-1213P
[107]Xiao-xun ZHANG, Zhen-shan CUI. Theoretical study of void closure in nonlinear plastic materials. Appl. Math. Mech,2009,30(5):631-642P
[108]Mata M, Casals O, Alcala J. The plastic zone size in indentation experiments:The analogy with the expansion of a spherical cavity. International Journal of Solids and Structures,2006,43:5994-6013P
[109]刘赵森,高建成.多孔泡沫金属抗侵彻能力研究.北京工业大学学报,2009,35(2)：156—161页
[110]周清强,孙锦山,王天书.动载荷下延性材料中微孔洞的增长模型.爆炸与冲击,2003,vo123(5)：415—419页
[111]张风国,崔亚平,秦承森.李勇.动载作用下材料的孔洞增长和成模.兵工学报,2004(6)：730—733页
[112]Gao X L et al. Two new expanding cavity models for indentation deformations of elastic strain-hardening materials. International Journal of Solids and Structures,2006,43: 2193-2208P
[113]Gao X L. Strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic-plastic material. Mech. Res. Comm.,2003,30(5):411-420P
[114]REN Jiusheng, CHENG Changjun. Static and dynamical cavitations for incompressible hyperelastic-plastic material. Journal of Shanghai University (English Edition),2006, 10(4):283-287P
[115]王勇,唐立强,吴国辉,边科.不可压缩组合球体的空穴和分叉.力学季刊,Vo129,No.3,2008：430—436页
[116]Galanov B A, Kartuzov V V, Ivanov S M. New analytical model of expansion of spherical cavity in brittle material based on the concepts of mechanics of compressible porous and powder materials. International Journal of Impact Engineering,2008,35 1522-1528P
[117]Leblond J B, Perrin G. Theoretical models for void coalescence in porous ductile solids. Int J Solids Struct,2001(32):5581-5604P
[118]Iqbal M A, Chakrabarti A, Beniwal S, Gupta N K.3D numerical simulations of sharp nosed projectile impact on ductile targets. International Journal of Impact Engineering, 2010(37):185-195 P
[119]Masri R, Durban D. Deep penetration analysis with dynamic cylindrical cavitation fields. International Journal of Impact Engineering,2009 (36):830-841P
[120]黄克智,萧纪美.材料的损伤断裂机理和宏微观力学理论.北京：清华大学出版社.1999：62—85P
[121]易顺民,朱珍德.裂隙岩体损伤力学导论.北京：科学出版社,2005：122—131页
[122]Collins I F. Elastic/Plastic models for Soils and Sands. Int. J. of Mechanical Sciences, 2005, Vol.47:493-508P.
[123]Collins I F, Hilder T. A Theoretical framework for construction elastic/plastic constitutive models for fraxial tests. Int. J. of Numerical and Analytical Methods in Geomechanics,2002, Vol.26:1313-1347P.
[124]张学言.岩土塑性力学基础.天津：天津大学出版社,2004：89—108页
[125]Sturmer G S, Schulz A and Wittig S. Life time prediction for ceramic gas turbine components.1991, ASME-Paper, No 91:96P
[126]Lade P V. Rock strength criteria-the theories and evidence, comprehensive rock engineering-principles, practice and projects, J A Hudson(ed), Pergamon Press, Oxford UK,1993:255-284P
[127]Andreev G E. Brittle Failure of Rock Material Test results and Constitutive Models, A A Balkema,1995
[128]Sheorey P R. Empirical rock failure criterion. A Balkema,1997
[129]Drucker D C and Prager W J. Soil mechanics and plastic analysis or limit design. Quat. of Appl. Math.,1952,10:157-165P
[130]Tschoegl N W. Failure Surfaces in Principal Stress Space. J. of Polymer Sci. Sym., 1971,32:239-267P
[131]Christensen R M. A comparative evaluation of three isotropic two property failure theories. J. Applied Mechanics,2006,73:852-859P
[132]Christensen R M. A two property yield, failure(fracture) criterion for homogeneous, isotropic materials. J. Egn. Mater. Technol,2004,126:45-52P
[133]唐立强,田德谟.一个压力敏感材料的本构方程.哈尔滨船舶工程学院学报,1994,15(1)：6—12页
[134]唐立强,谭英杰,郑贵.泥岩本构方程的研究,哈尔滨工程大学学报,2004,24,(1)：92—97页
[135]Chandler H W. Homogeneous and localized deformation in granular materials:A mechanistic model. Int. J. Eng. Sci.,1990,28(8):719-134P
[136]Wilson C D. A critical reexamination of classical metal plasticity. J. Applied Mechanics,2002,69:63-68P
[137]Deshpande V S, Fleck N A. Isotropic constitutive models for metallic foams. Journal of the Mechanics and Physics of Solids,2000,48:1253-1283P
[138]Andrews E W, Sanders W, Gibson L J. Compressive and tensile behavior of aluminum foams. Materials Science and Engineering A,1999,270(2):113-124P
[139]Gioux G, McCormack T M, Gibson L J. Failure of aluminum foams under multiaxial loads. International Journal of Mechanical Sciences,2000,42:1097-1117P
[140]Chen H J and Tang L Q. Crack—tip asymptotic field in porous materials, China: Mesomechanics 2000,2000,1:305-311P
[141]Liqiang Tang, Xuemei Yu, Xiaoying Zhang. Yield Criteria for Pressure-sensitive Materials based on Meso-mechanical Analysis. Key Engineering Materials Vols,417-418(2010):893-896P
[142]王仁,黄文彬,黄筑平.塑性力学引论.北京：北京大学出版社,1992：113—139页
[143]孙秋华,蔡艳红,唐立强.非线性损伤材料Ⅲ型裂纹尖端场.哈尔滨工程大学学报,2003,24,(2)：192—197页
[144]Costin L S. Time-dependent damage and creep of brittle rock. Damage Mechanics and Continuum Modeling. Stubbs N, Krajcinovic Deds, New York:ASCE,1985:25-38P
[145]于雪梅,唐立强,王堃.内压作用下球形孔洞膨胀损伤力学分析.哈尔滨工程大学学报,2010,No.8：1034-1038页
[146]Yu Xuemei, Tang Liqiang, Yang Yong. Static solution on four-region spherical cavity expansion model in pressure sensitive medium. Journal of Harbin Institute of Technology,2010,17(3):99-102P
[147]王自强,陈少华.高等断裂力学.北京：科学出版社,2009：14—39页
[148]Krajcinovic D and Fonseka G U. The continuous damage theory of brittle materials. Part 1:general theory. J. Appl.Mech,1981,48:809-815P
[149]于骁中.岩石和混凝土断裂力学.中南工业大学出版社,1991：417—510页
[150]唐立强,于雪梅,吴国辉.压力敏感性材料球形孔洞动态扩展问题研究.哈尔滨工业大学学报,2010,42(4)：1569—1572页.
[151]Durban D, Masri R. Dynamic spherical cavity expansion in a pressure sensitive elastoplastic medium. Int. J. Solids Struct.,2004,41(20):5697-5716P
[152]Carroll M M and Holt A C. Static and dynamic pore-collapse relation for ductile porous materials. Journal of Applies Physics,1972,43:1626-1636P
[153]任九生,程昌钧.受内压热超弹性球壳的不稳定性.上海大学学报(自然科学版),Vol.13 No.6,2007：732—740页
[154]Srinirasa A R. Large deformation plasticity and the effect. Int. J. Plasticity.,2001, 17(9):1189-1214P
[155]Maohong Yu, Guiyun Xia, Vladimir A Kolupaev. Basic characteristics and development of yield criteria for geomaterials. Journal of Rock Mechanics and Geotechnical Engineering,2009,1(1):71-88P
[156]吕霁,崔颖辉,刘佳,鲁海.岩石力学强度理论的研究现状分析.北方工业大学学报,Vol.22,No.1,2010：73—78页
[157]Cohen T, Masri R, Durban D. Analysis of circular hole expansion with generalized yield criteria. International Journal of Solids and Structures, Vol.46,2009:3643-3650P
[158]Xiao-xun ZHANG, Zhen-shan CUI. Theoretical study of void closure in nonlinear plastic materials. Appl. Math. Mech.-Engel. Ed., Vol.30, No.5,2009:631-642P
[159]Galanov B A, Kartuzov V V, Ivanov S M. New analytical model of expansion of spherical cavity in brittle material based on the concepts of mechanics of compressible porous and powder materials. International Journal of Impact Engineering,2008, Vol.35:1522-1528P
[160]Wei X Y, Zhao Z Y, Gu J. Numerical simulation s of rock mass damage induced by underground explosion. International Journal of Rock Mechanics & Mining Sciences, 2009, Vol.46:1206-1213P
[161]Ogden R W. Saccomandi G. Introducing microscopic information into constitutive equations for arterial walls. Biomechanics and Modeling in Methane-biology,2007, 6(5):333-344P
[2]谢和平,陈忠辉.岩石力学.北京：科学出版社,2004：46—64页
[3]李翼祺,马素贞著.爆炸力学.北京：科学出版社,1992：356—417页.
[4]杨桂通.土动力学.北京：中国建材工业出版社,2000：92—134页.
[5]张晓春等.岩体裂纹演化及其力学特性的研究进展.力学进展.1999,29(1)：97—104页
[6]陈惠发.土木工程材料的本构关系—塑性与建模.武汉：华中科技大学出版社,2001：212—267页
[7]Gibson L J, Ashby M F. Cellular Solids:Structure and Properties[M].Oxford:Pergamon Press,1997:32-50P
[8]Gibson L J. Mechanic behavior of metallic foam. Annual Review of Materials Science,2000,30:191-227P
[9]Bishop R F, Hill R, Mott N F. The theory of indentation and hardness. Test. Proc. Phys. Soc.1945,57(3):147-159P.
[10]Taylor G I. The formation and enlargement of a circular hole in a thin plastic sheet. Quarterly. J. Mech. and Appl. Math.,1948,1:103-124P
[11]Hill R. The Mathematical Theory of Plasticity. London. Oxford University Press,1950: 89-101P
[12]Hopkins H G. Dynamic expansion of spherical cavities in metal. Progress in Solid Mechanics, Edited by Sneddon I N, Hill R. North-Holland, Amsterdam,1960.
[13]Macek R W, Duffey T A. Finite cavity expansion method for near-surface effects and layering during Earth penetration. International Journal of Impact Engineering,2000, 24(3):239-258P
[14]Rosenberg Z, Dekel E. A numerical study of the cavity expansion process and its application to long-rod penetration mechanics. International Journal of Impact Engineering. No.35,2008:147-154P
[15]俞茂宏.强度理论百年总结.力学进展.2004,34(4)：529—560页
[16]Jianchun Li, Guowei Ma, Maohong Yu. Penetration analysis for geo-material based on unified strength criterion. International Journal of Impact Engineering.2008,35:1154 -1163P
[17]Maohong Yu, Guiyun Xia, V A Kolupaev. Basic characteristics and development of yield criteria for geomaterials. Journal of Rock Mechanics and Geotechnical Engineering,2009,1(1):71-88P
[18]余寿文,冯西桥.损伤力学.北京：清华大学出版社,1997：78—114页
[19]Mura T. Miero mechanics of Defects in Solids, Martinus Nijhoff Publishers,1987: 53-61P
[20]王自强,段祝平.塑性细观力学.北京：科学出版社,1995：242—275页
[21]杨卫.宏细观断裂力学.北京：国防工业出版社,1995：134—288页
[22]卡恰诺夫,连续介质损伤力学.哈尔滨：哈尔滨工业大学出社,1989：1—10P,123—125页
[23]Rabotnov Y N. Creep problems in structural members. North-Holland, Amsterdam,1969.
[24]Lemaitre J, Chaboche J L. Mechanics of Solid Materials, Cambridge:Cambridge University Press,1988:317-329P
[25]Lemaitre J. A course on damage mechanics. Springer-Verlag, Belin.1992.
[26]Hult J and Broberg H. Creep rupture under cyclic lading. Proc. of the second Bulgarran Congress on Mechanics,1976:263-272P
[27]Leckie F A and Hayhurst D R. Constitutive equation of creep rupture. Acta Meta, 1977(25):1059-1076P
[28]Murakami S. Progress of damage mechanics. JSME Inter. Journal.1987,30, No.263: 701-710P
[29]Gurson A L. Continuum theory of ductile rupture by the void nucleation and growth. J. Engng. Mater. Technology,1977,99:234-245P
[30]McClintock F A. A criterion for ductile fracture by the growth of holes. J. Appl. Mech.1968,35:363-371P
[31]Rice I R and Tracey D M. On the ductile enlargement of voids in triaxil stress fields. J. Mech. Phys. Solids,1969,17:201-227P.
[32]Lee J H, Chung J. Yield functions and flow rules for porous pressure-dependent strain-hardening polymeric materials, ASME, Journal of Applied Mechanics,2000, 67:288-297P
[33]Xia L and Shih C F. Ductile crack growth-I, A numerical study using computational cells with micro-structurally based length scales. J. Mech. Phys. Solids.,1995(43): 233-259P
[34]Xia L, Shih C F. Ductile crack growth-Ⅱ, A numerical study using computational cells with micro-structurally based length scales. J. Mech Phys Solids,1995(43):1953-1981P
[35]Tvergaard V, Needleman A. Effects of non-local damage in porous plastic solids. International Journal of Solids and Structures,1995(32):1063-1077P
[36]Needleman A, Tvergaard V. Micromechanical analysis of the ductile-brittle transition. Engineering Fracture Mechanics,1999(62):317-338P
[37]Biglari F R, Nikbin K M, O'Dowd N P. Numerical simulation of crack growth based on void volume fraction. Proceedings of the 10th annual mechanical engineering conference, ISME2002, Tehran Iran,2002,5:492-496P
[38]Tvergaard V, Hutchinson J W. Two mechanisms of ductile fracture:void by void growth versus multiple void interactions. Int. J. Solids Structures,2002,39:3581-3597P.
[39]Tvergaard V, Niordson C. Nonlocal plasticity effects on interaction of different size voids. International Journal of Plasticity,2004,20(1):107-120P
[40]Tvergaard V, Needleman A. Three-dimensional micro-structural effects on plane strain ductile crack growth. Int J Solids Struct,2006,43(20):6165-6179P
[41]董杰,李永池,陈学东.一种新型微孔洞损伤模型.高压物理学报.2007,21,(4)：414—418页
[42]董杰,李永池,陈学东.微孔洞唯象损伤力学模型及其应用.爆炸与冲击.2008,8(5)：443-447页
[43]王自强,秦嘉亮.含空洞非线性材料的本构势和空洞扩展率.固体力学学报,1989,10(2)：127—141页
[44]黄筑平,孙立志.韧性材料中的一个新的动态孔洞增长模型.中国科学A辑,1992(9)：960—969页
[45]Ashby M F, Hallam S D. The failure of brittle solids containing small cracks under compressive stress states. Acta Metall.1986,34:497-510P.
[46]Horii H S, Nemat-Nasser. Overall moduli of solids with microcracks:load-induced anisotropy. J. Mech. Phys.Solids.1983,31:155-171P.
[47]Horii H S, Nemat-Nasser. Brittle fracture in compression:splitting, faulting and brittle-ductile transition. Phil. Trans. Roy. Soc, London:1986, A:337-374P.
[48]Budiansky B, O'Connell R J. Elastic module of a cracked solids. Int. J. Solids Structures,1976,12:81-97P.
[49]高玉臣,朱葳.含微裂纹材料的损伤理论.力学学报,1987,19：541—54页.
[50]唐立强,陈会军,姜民政.海冰板的环形屈曲和压缩破坏.海洋工程,2001,19,(4)：29—33页
[51]赵吉坤,张子明,仝兴华,薛世峰.岩石宏细观弹塑性损伤破坏对比研究.中国石油大学学报(自然科学版),中国石油大学学报(自然科学版)2007,31：78—84页
[52]黄筑平,杨黎明,潘客麟.材料的动态损伤和失效.力学进展,1993,23(4)：433—467页
[53]Butterfield R and Banerjee P K. The effects of pore water pressures on the ultimate bearing capacity of driven piles. In:Proc 2nd south East Asian regional Soil Mech Found Engrg, Tykyo,1970,385-394P
[54]Vesic A S. Expansion of Cavity in an Infinite Soil Mass, J.S.M. F.E. ASCE,1972, 98(3):265-289P
[55]Carter J P, Booker J R and Yeung S K. Cavity Expansion in Cohesive Frictional Soil. Geotechnique 1986,36(3):349-353P
[56]Randoph M F, Carter L P, Wroth C P. Driven Piles in Clay-the Effects of Installation and Subsequent Consolidation. Geotechnique,1979,29(4):361-393P
[57]Salgado R, Mitchel J K, Jamiolkowski. Cavity expansion and penetration resistance in sand. Jour Geotech Geoenvior Engng, ASCE,1997,123(4):344-357P
[58]Chow Y K and Thec I. A Theoretical Study of Pile Heave. Geotechnique,1990,40(1): 1-14P
[59]Mantaras F M, Schnaid F. Cylindrical cavity expansion in dilatants cohesive—frictional materials. Geotechnique,2002,52(5):337-348P
[60]Yu H S, Houlsby G T. Finite cavity expansion in dilatant soils:loading analysis. Geotechnique,1991,41(2):173-183P
[61]Cao L F, Teh C I, Chang M F. Undrained cavity expansion in modified Cam clay Ⅰ: theoretical analysis[J]. Geotechnique,2001,51(4):323-334P
[62]Chang M F, Teh C I, Cao L F. Undrained cavity expansion in modified Cam-clay Ⅱ: application to the interpretation of the piezocone test. Geotechnique,2001,51(4): 335-350P
[63]李月健,陈云敏等,土体内空穴球形扩张问题的一般解及应用.土木工程学报,2002, 35(1)：93-98页
[64]叶建忠,周健.砂土中桩的刺入变形与端阻力发挥的初步研究.建筑结构.2007,Vol37,No.8：P88—90页
[65]刘俊伟,张明义,赵洪福,王静静.基于球孔扩张理论和侧阻力退化效应压桩力计算模拟.岩土力学,Vol.30,No.4,2009：1181—1185页
[66]李赞成,胡功笠,梁济丰,张韬.考虑动态参数的混凝土侵彻工程计算.力学与实践,Vol.31,No.6,2009：42—45页
[67]罗战友,夏建中,龚晓南.不同拉压模量及软化特性材料的柱形孔扩张问题的统一解.工程力学,2008,25(9)：79—84页
[68]汪鹏程,张红亚.弹塑脆性岩土材料中球形孔大应变扩张分析.安徽建筑工业学院学报(自然科学版),2008,16(2)：18—22页
[69]Zhou Aizhao, Lu Tinghao. Elasto—plastic constitutive model of soil—structure interface in consideration of strain softening and dilation. Acta Mechanics Solid Sinica, 2009,22(2):171-179P
[70]胡士兵,王金昌,朱向荣.线性软化土体中球孔扩张问题的解析解.浙江大学学报(工学版),2007,41(9)：1503—1507页
[71]林政,陈云敏,陈仁朋.球形空腔固结解析解及其在地基原位固结系数测试中的应用.岩石力学与工程学报,2005,24(2)：5863—5867页
[72]Feiberger W. A problem in dynamic plasticity:the enlargement of a circular of ho 1 e in a flat sheet. Proc. of the Cambridge Philosophical Society,1952, V01(48):135 P
[73]Forrestal M J, and Luk V K. Dynamic Spherical Cavity—Expansion in a Compressible Elastic-Plastic Solid, J. Appl. Mech.,1988,55:275-279P
[74]Luk V K, Amos D E. Dynamic cylindrical cavity expansion of compressible strain-hardening materials. J. Appl. Mech.,1991,58(2):334-340P
[75]Forrestal M J, Brar N S, LukVK. Penetration of strain— hardening targets with rigid spherical-nose rods. J. Appl. Mech.,1991,58:7-10P
[76]Forrestal M J, Tzou D Y, Askari E, Longcope D B. Penetration into ductile metal targets with rigid spherical—nose rods. Int. J. Impact Eng.,1995,16(5-6):699-710P
[77]Forrestal M J and Tzou D Y. A spherical cavity expansion penetration model for concrete targets. International Journal of Plasticity,1998,14(1-3):173-191P
[78]Satapathy S. Dynamic spherical cavity expansion in brittle ceram -ics. International Journal of Solids and Structures,2001(38):5833-5845P
[79]Satapathy S and Blss S J. Cavity expansion resistance of in brittle materials obeying a two curve pressure-shear behavior. J. Appl. Phys.2000,88(7):4004-4012P
[80]Rosenberg Z, Dekel E. A numerical study of the cavity expansion process and its application to long-rod penetration. International Journal of Impact Engineering,2008, 35:147-154P
[81]Masri R, Durban D. Dynamic spherical cavity expansion in an elastoplastic compressible mises solid. J. Appl. Mech.,2005,72(6):887-898P
[82]Masri R, Durban D. Dynamic cylindrical cavity expansion in an incompressible elastoplastic medium. Acta Mech,2006,181(1-2):105-123P
[83]Masri R, Durban D. Cylindrical cavity expansion in compressible mises and tresca solids. European Journal of Mechanics A/Solids 2007,26(4):712-727P
[84]Narasimhan R. Analysis of indentation of pressure sensitive plastic solids using the expanding cavity model. Mech. Mater.2004,36:633-645P
[85]何涛,文鹤鸣.球形弹对金属靶板侵彻问题的数值模拟.爆炸与冲击,Vol.26,No.5,2006,9：456—461页
[86]蒋志刚,曾首义,周建平.中等厚度金属靶板的三阶段贯穿模型.兵工学报,2007,Vol28,No9：1046—1052页
[87]吴昊,方秦,龚自明,相恒波.应用改进的双剪强度理论分析岩石靶体的弹体侵彻深度.工程力学,2009,Vol.26,No.8：216—222页
[88]Frew D J, Forrestal M J, Cargile J D. The effect of concrete target diameter on projectile deceleration and penetration depth. International Journal of Impact Engineering,2006,32:1584-1594P
[89]韩丽,高世桥,李明辉,刘海鹏.弹丸垂直贯穿混凝土靶的数值研究.北京理工大学学报,2006,26(11)：953—956页
[90]Kogat I, Eslion I. Elastic-plastic contact analysis of a sphere and a rigid flat. J. Appl. Mech.,2002,69(5):657-662 P
[91]Fischer-Cripps A C. Elastic-plastic behavior in materials loaded with a spherical indenter. J. Mater. Sci.,1997,32(3):727-736 P
[92]Mata M, Alcala J. Mechanical property evaluation through sharp indentation in elastoplastic and fully plastic contact regimes. J. Mater. Res.,2003,18(9): 1705-1709P
[93]Mata M, Anglada M and Alcala J. A hardness equation for sharp indentation of elastic-power-law strain-hardening materials. Philos. Mag.2002,82(10):1831-1839P
[94]Mesarovic S D, Fleck N A. Spherical indentation of elastic-plastic solids. Proc. R. Soc. Lond.,1999,455:2707-2728P
[95]Park Y J, Pharr G M. Nanoindentation with spherical indenters:finite element studies of deformation in the elastic-plastic transition regime. Thin Solid Films,2004, 447-448(30):246-250P
[96]Wei Y and Hutchinson J W. Hardness trends in micron scale indentation. J. Mech. Phys. Solids,2003,51,2037-2056P.
[97]Johnson K L. Contact Mechanics. Cambridge:Cambridge University Press,1985: 175P
[98]Chadwick P. The quasi-static expansion of a spherical cavity in metals and ideal soils. Quart. J. Mech. Appl. Math.,1959,12(1):52-71P
[99]Durban D, Baruch M. On the problem of a spherical cavity in an infinite elasto-plastic medium. J. Appl. Mech.,1976,43:633-638P
[100]Durban D, Kubi M. A general solution for the pressurized elastoplastic tube. J. Appl. Mech.,1992,59:20-26P
[101]Durban D, Papanastasiou P. Cylindrical cavity expansion and contraction in pressure sensitive geomaterials. Acta Mech.,1997,122(1-4):99-122P
[102]Brignoni D, Ludiero F. The quasi-static finite cavity expansion in a non-standard elasto-plastic medium. Int. J. Mech. Sic.,1989,31(11-12):825-837P
[103]Jin Ming. Huang kefu. Jike wu. A study of the cafastrophe and the caritation for a spherical cavity in Hook's material with 1/2 poision'ratio. Applied Meth Mech,1999 20(8):928-935P
[104]程昌钧,聂波.幂硬化材料和超弹性材料含球体中空穴的动态生成.固体力学学报,2005,vol26(3)：273—279页
[105]任九生,程昌钧.受内压热超弹性球壳的不稳定性.上海大学学报(自然科学版),2007,13(6)：732—740页
[106]Wei X Y, Zhao Z Y, Gu J. Numerical simulations of rock mass damage induced by underground explosion. International Journal of Rock Mechanics & Mining Sciences, 2009,46:1206-1213P
[107]Xiao-xun ZHANG, Zhen-shan CUI. Theoretical study of void closure in nonlinear plastic materials. Appl. Math. Mech,2009,30(5):631-642P
[108]Mata M, Casals O, Alcala J. The plastic zone size in indentation experiments:The analogy with the expansion of a spherical cavity. International Journal of Solids and Structures,2006,43:5994-6013P
[109]刘赵森,高建成.多孔泡沫金属抗侵彻能力研究.北京工业大学学报,2009,35(2)：156—161页
[110]周清强,孙锦山,王天书.动载荷下延性材料中微孔洞的增长模型.爆炸与冲击,2003,vo123(5)：415—419页
[111]张风国,崔亚平,秦承森.李勇.动载作用下材料的孔洞增长和成模.兵工学报,2004(6)：730—733页
[112]Gao X L et al. Two new expanding cavity models for indentation deformations of elastic strain-hardening materials. International Journal of Solids and Structures,2006,43: 2193-2208P
[113]Gao X L. Strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic-plastic material. Mech. Res. Comm.,2003,30(5):411-420P
[114]REN Jiusheng, CHENG Changjun. Static and dynamical cavitations for incompressible hyperelastic-plastic material. Journal of Shanghai University (English Edition),2006, 10(4):283-287P
[115]王勇,唐立强,吴国辉,边科.不可压缩组合球体的空穴和分叉.力学季刊,Vo129,No.3,2008：430—436页
[116]Galanov B A, Kartuzov V V, Ivanov S M. New analytical model of expansion of spherical cavity in brittle material based on the concepts of mechanics of compressible porous and powder materials. International Journal of Impact Engineering,2008,35 1522-1528P
[117]Leblond J B, Perrin G. Theoretical models for void coalescence in porous ductile solids. Int J Solids Struct,2001(32):5581-5604P
[118]Iqbal M A, Chakrabarti A, Beniwal S, Gupta N K.3D numerical simulations of sharp nosed projectile impact on ductile targets. International Journal of Impact Engineering, 2010(37):185-195 P
[119]Masri R, Durban D. Deep penetration analysis with dynamic cylindrical cavitation fields. International Journal of Impact Engineering,2009 (36):830-841P
[120]黄克智,萧纪美.材料的损伤断裂机理和宏微观力学理论.北京：清华大学出版社.1999：62—85P
[121]易顺民,朱珍德.裂隙岩体损伤力学导论.北京：科学出版社,2005：122—131页
[122]Collins I F. Elastic/Plastic models for Soils and Sands. Int. J. of Mechanical Sciences, 2005, Vol.47:493-508P.
[123]Collins I F, Hilder T. A Theoretical framework for construction elastic/plastic constitutive models for fraxial tests. Int. J. of Numerical and Analytical Methods in Geomechanics,2002, Vol.26:1313-1347P.
[124]张学言.岩土塑性力学基础.天津：天津大学出版社,2004：89—108页
[125]Sturmer G S, Schulz A and Wittig S. Life time prediction for ceramic gas turbine components.1991, ASME-Paper, No 91:96P
[126]Lade P V. Rock strength criteria-the theories and evidence, comprehensive rock engineering-principles, practice and projects, J A Hudson(ed), Pergamon Press, Oxford UK,1993:255-284P
[127]Andreev G E. Brittle Failure of Rock Material Test results and Constitutive Models, A A Balkema,1995
[128]Sheorey P R. Empirical rock failure criterion. A Balkema,1997
[129]Drucker D C and Prager W J. Soil mechanics and plastic analysis or limit design. Quat. of Appl. Math.,1952,10:157-165P
[130]Tschoegl N W. Failure Surfaces in Principal Stress Space. J. of Polymer Sci. Sym., 1971,32:239-267P
[131]Christensen R M. A comparative evaluation of three isotropic two property failure theories. J. Applied Mechanics,2006,73:852-859P
[132]Christensen R M. A two property yield, failure(fracture) criterion for homogeneous, isotropic materials. J. Egn. Mater. Technol,2004,126:45-52P
[133]唐立强,田德谟.一个压力敏感材料的本构方程.哈尔滨船舶工程学院学报,1994,15(1)：6—12页
[134]唐立强,谭英杰,郑贵.泥岩本构方程的研究,哈尔滨工程大学学报,2004,24,(1)：92—97页
[135]Chandler H W. Homogeneous and localized deformation in granular materials:A mechanistic model. Int. J. Eng. Sci.,1990,28(8):719-134P
[136]Wilson C D. A critical reexamination of classical metal plasticity. J. Applied Mechanics,2002,69:63-68P
[137]Deshpande V S, Fleck N A. Isotropic constitutive models for metallic foams. Journal of the Mechanics and Physics of Solids,2000,48:1253-1283P
[138]Andrews E W, Sanders W, Gibson L J. Compressive and tensile behavior of aluminum foams. Materials Science and Engineering A,1999,270(2):113-124P
[139]Gioux G, McCormack T M, Gibson L J. Failure of aluminum foams under multiaxial loads. International Journal of Mechanical Sciences,2000,42:1097-1117P
[140]Chen H J and Tang L Q. Crack—tip asymptotic field in porous materials, China: Mesomechanics 2000,2000,1:305-311P
[141]Liqiang Tang, Xuemei Yu, Xiaoying Zhang. Yield Criteria for Pressure-sensitive Materials based on Meso-mechanical Analysis. Key Engineering Materials Vols,417-418(2010):893-896P
[142]王仁,黄文彬,黄筑平.塑性力学引论.北京：北京大学出版社,1992：113—139页
[143]孙秋华,蔡艳红,唐立强.非线性损伤材料Ⅲ型裂纹尖端场.哈尔滨工程大学学报,2003,24,(2)：192—197页
[144]Costin L S. Time-dependent damage and creep of brittle rock. Damage Mechanics and Continuum Modeling. Stubbs N, Krajcinovic Deds, New York:ASCE,1985:25-38P
[145]于雪梅,唐立强,王堃.内压作用下球形孔洞膨胀损伤力学分析.哈尔滨工程大学学报,2010,No.8：1034-1038页
[146]Yu Xuemei, Tang Liqiang, Yang Yong. Static solution on four-region spherical cavity expansion model in pressure sensitive medium. Journal of Harbin Institute of Technology,2010,17(3):99-102P
[147]王自强,陈少华.高等断裂力学.北京：科学出版社,2009：14—39页
[148]Krajcinovic D and Fonseka G U. The continuous damage theory of brittle materials. Part 1:general theory. J. Appl.Mech,1981,48:809-815P
[149]于骁中.岩石和混凝土断裂力学.中南工业大学出版社,1991：417—510页
[150]唐立强,于雪梅,吴国辉.压力敏感性材料球形孔洞动态扩展问题研究.哈尔滨工业大学学报,2010,42(4)：1569—1572页.
[151]Durban D, Masri R. Dynamic spherical cavity expansion in a pressure sensitive elastoplastic medium. Int. J. Solids Struct.,2004,41(20):5697-5716P
[152]Carroll M M and Holt A C. Static and dynamic pore-collapse relation for ductile porous materials. Journal of Applies Physics,1972,43:1626-1636P
[153]任九生,程昌钧.受内压热超弹性球壳的不稳定性.上海大学学报(自然科学版),Vol.13 No.6,2007：732—740页
[154]Srinirasa A R. Large deformation plasticity and the effect. Int. J. Plasticity.,2001, 17(9):1189-1214P
[155]Maohong Yu, Guiyun Xia, Vladimir A Kolupaev. Basic characteristics and development of yield criteria for geomaterials. Journal of Rock Mechanics and Geotechnical Engineering,2009,1(1):71-88P
[156]吕霁,崔颖辉,刘佳,鲁海.岩石力学强度理论的研究现状分析.北方工业大学学报,Vol.22,No.1,2010：73—78页
[157]Cohen T, Masri R, Durban D. Analysis of circular hole expansion with generalized yield criteria. International Journal of Solids and Structures, Vol.46,2009:3643-3650P
[158]Xiao-xun ZHANG, Zhen-shan CUI. Theoretical study of void closure in nonlinear plastic materials. Appl. Math. Mech.-Engel. Ed., Vol.30, No.5,2009:631-642P
[159]Galanov B A, Kartuzov V V, Ivanov S M. New analytical model of expansion of spherical cavity in brittle material based on the concepts of mechanics of compressible porous and powder materials. International Journal of Impact Engineering,2008, Vol.35:1522-1528P
[160]Wei X Y, Zhao Z Y, Gu J. Numerical simulation s of rock mass damage induced by underground explosion. International Journal of Rock Mechanics & Mining Sciences, 2009, Vol.46:1206-1213P
[161]Ogden R W. Saccomandi G. Introducing microscopic information into constitutive equations for arterial walls. Biomechanics and Modeling in Methane-biology,2007, 6(5):333-344P