基于能量等效原理的应变局部化分析:Ⅱ.有限元解法
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摘要
以非局部塑性理论为基础,应用状态空间理论,通过局部和非局部两个状态空间的塑性能量耗散率等效原理,提出了一种求解应变局部化问题的新方法,以得到与网格无关的数值解.针对二维问题的屈服函数和流动法则导出了求解非局部内变量的一般方程,并提出了在有限元环境中求解应变局部化问题的应力更新算法.为了验证所提出的方法,对1个一维拉杆和3个二维平面应变加载试件进行了有限元分析.数值结果表明,塑性应变的分布和载荷-位移曲线都随着网格的变小而稳定地收敛,应变局部化区域的尺寸只与材料内尺度有关,而对有限元网格的大小不敏感.对于一维问题,当有限元网格尺寸减小时,数值解收敛于解析解.对于二维剪切带局部化问题,数值解随着网格尺寸的减小而稳定地向唯一解收敛.当网格尺寸减小时,剪切带的宽度和方向基本上没有变化.而且得到的塑性应变分布和网格变形是平滑的.这说明,所提方法可以克服经典连续介质力学模型导致的网格相关性问题,从而获得具有物理意义的客观解.此模型只需要单元之间的位移插值函数具有C~0连续性,因而容易在现有的有限元程序中实现而无需对程序作大的修改.
Founded on the nonlocal plasticity and the state space theories, a new approach is proposed to find the meshindependent solution of the strain localization problems by equating the rates of plastic energy dissipation in the local and nonlocal state spaces. Following the previous paper by the authors, general formulas are developed for the solution of the nonlocal internal variables in the two-and more than two-dimensional problems. A stress updating algorithm is proposed to integrate the rate form constitutive equations in the finite element context. To verify the proposed approach,a one-dimensional model problem and three two-dimensional plane strain problems are solved numerically by the finite element method. Numerical results show that the plastic strain distributions and the load-displacement curves stably converge with refinement of the finite element mesh. The size of the localization zone depends only on the internal length scale and is insensitive to the mesh size. For the one-dimensional problem, numerical solutions converge to the analytical ones. For the two-dimensional problems, although no analytical solutions are available, the numerical solutions converge toward the unique ones. The width and the inclination are almost not changed as the mesh size is reduced. Also, the distribution of the plastic strains and the deformation patterns are smooth in the entire domain. A slope stability problem and a plane strain test of a coal specimen are also solved numerically to demonstrate the robustness of the proposed approach. It is well shown that the proposed approach can overcome the drawbacks of the classical continuum theory and lead to physically meaningful, mesh-independent solution of strain softening problems. Because only C0 continuity is needed between element boundaries, the proposed approach is easy to be incorporated into the existing finite element code without substantial modification.
Founded on the nonlocal plasticity and the state space theories, a new approach is proposed to find the meshindependent solution of the strain localization problems by equating the rates of plastic energy dissipation in the local and nonlocal state spaces. Following the previous paper by the authors, general formulas are developed for the solution of the nonlocal internal variables in the two-and more than two-dimensional problems. A stress updating algorithm is proposed to integrate the rate form constitutive equations in the finite element context. To verify the proposed approach,a one-dimensional model problem and three two-dimensional plane strain problems are solved numerically by the finite element method. Numerical results show that the plastic strain distributions and the load-displacement curves stably converge with refinement of the finite element mesh. The size of the localization zone depends only on the internal length scale and is insensitive to the mesh size. For the one-dimensional problem, numerical solutions converge to the analytical ones. For the two-dimensional problems, although no analytical solutions are available, the numerical solutions converge toward the unique ones. The width and the inclination are almost not changed as the mesh size is reduced. Also, the distribution of the plastic strains and the deformation patterns are smooth in the entire domain. A slope stability problem and a plane strain test of a coal specimen are also solved numerically to demonstrate the robustness of the proposed approach. It is well shown that the proposed approach can overcome the drawbacks of the classical continuum theory and lead to physically meaningful, mesh-independent solution of strain softening problems. Because only C0 continuity is needed between element boundaries, the proposed approach is easy to be incorporated into the existing finite element code without substantial modification.
引文
1黄茂松.土体稳定与承载特性的分析方法.岩土工程学报,2016,38 (1):1-34(Huang Maosong.Analysis methods for stability and bearing capacity of soils.Chinese Journal of Geotechnical Engineering,2016,38(1):1-34(in Chinese))
2 钱建固,吕玺琳,黄茂松.平面应变状态下土体的软化特性与本构模拟.岩土力学,2009,30(3):617-622(Qian Jiangu,L¨u Xilin,Huang Maosong.Softening characteristics of soils and constitutive modeling under plane strain condition.Rock and Soil Mechanics,2009,30(3):617-622(in Chinese))
3 徐连民,王天竹,祁德庆等.岩土中的剪切带局部化问题研究:回顾与展望.力学季刊,2004,25(4):484-489(Xu Lianmin,Wang Tianzhu,Qi Deqing,et al.Study on geotechnical shear band localization:retrospect and prospect.Chinese Quarterly of Mechanics,2004,25(4):484-489(in Chinese))
4 张启辉,赵锡宏.粘性土局部化剪切带变形的机理研究.岩土力学,2002,23(1):31-35(Zhang Qihui,Zhao Xihong.Study on mechanism of localized shear band formation in clay.Rock and Soil Mechanics,2002,23(1):31-35(in Chinese))
5 王杰,李世海,张青波.基于单元破裂的岩石裂纹扩展模拟方法.力学学报,2015,47(1):105-118(Wang Jie,Li Sihai,Zhang Qingbo.Simulation of crack propagation of rock based on splitting elements.Chinese Journal of Theoretical and Applied Mechanics,2015,47(1):105 -118(in Chinese))
6 李宏儒,胡再强,冯飞等.结构性黄土二元介质本构模型在局部化剪切带中的应用.岩土力学,2012,33(9):2803-2844(Li Hongru,Hu Zaiqiang,Feng Fei,et al.Application of structural loess binarymedium model to localization shear band.Rock and Soil Mechanics,2012,33(9):2803-2844(in Chinese))
7 黄茂松,贾苍琴,钱建固.岩土材料应变局部化的有限元分析方法.计算力学学报,2007,24(4):465-471(Huang Maosong,Jia Cangqin,Qian Jiangu.Strain localization problems in geomaterials using finite elements.Chinese Journal of Computational Mechanics,2007,24(4):465-471(in Chinese))
8 邵龙潭,刘港,郭晓霞.三轴试样破坏后应变局部化影响的实验研究.岩土工程学报,2016,38(3):385-394(Shao Longtan,Liu Gang,Guo Xiaoxia.Effects of strain localization of triaxial samples in post-failure state.Chinese Journal of Rock Mechanics and Engineering,2016,38(3):385-394(in Chinese))
9 李学丰,黄茂松,钱建固.基于非共轴理论的各向异性砂土应变局部化分析.工程力学,2014,31(3):205-246(Li Xuefeng,Huang Maosong,Qian Jiangu.Strain localization analysis of anisotropic sands based on non-coaxial theory.Engineering Mechanics,2014,31 (3):205-246(in Chinese))
10 王水林,郑宏,刘泉声等.应变软化岩体分析原理及其应用,岩土力学,2014,35(3):609-630(Wang Shuilin,Zheng Hong,Liu Quansheng,et al.Principle of analysis of strain-softening rock mass and its application.Rock and Soil Mechanics,2014,35(3):609-630(in Chinese))
11 Read HE,Hegemier GA.Strain softening of rock,soil and concrete—a review article.Mechanics and Materials,1984,3(3):271-294
12 Vardoulakis I.Shear band inclination and shear modulus of sand in biaxial test.International Journal for Numerical and Analytical Methods in Geomechanics,1980,4(2):103-119
13 Jir′asek M,Ba?zant ZP.Inelastic analysis of structures.Chichester,England:John Wiley&Sons,2002
14 Pietruszczak ST,Mr′oz Z.Finite element analysis of deformation of strain-softening materials.International Journal for Numerical Methods in Engineering,1981,17(3):327-334
15 Ortiz M,Leroy Y,Needleman A.A finite element method for localized failure analysis.Computer Methods in Applied Mechanics and Engineering,1987,61(2):189-214
16 de Borst R,Sluys LJ,M¨uhlhaus H-B,et al.Fundamental issues in finite element analyses of localization of deformation.Engineering Computations,1993,10(2):99-121
17 Simo JC.Strain softening and dissipation:a unification of approaches//Mazars J,Ba?zant JP,eds.Cracking and Damage:strain localization and size effect.London and New York:Elsevier Applied Science,1988:440-461
18 de Borst R,M¨uhlhaus H-B.Gradient-dependent plasticity:formulation and algorithm aspects.International Journal for Numerical Methods in Engineering,1992,35(3):521-539
19 Luzio GD,Ba?zant ZP.Spectral analysis of localization in nonlocal and over-nonlocal materials with softening plasticity or damage.International Journal of Solids and Structures,2005,42(23):6071-6100
20 Vermeer PA,Brinkgreve RBJ.A new effective non-local strain measure for softening plasticity//Chambon V,Desrues J,Vardoulakis I,eds.Localisation and Bifurcation Theory for Soils and Rocks.Rotterdam,Netherland:A.A.Balkema,1994
21 Stromberg L,Ristinmaa M.FE-formulation of a nonlocal plasticity theory.Computer Methods in Applied Mechanics and Engineering,1996,136(1-2):127-144
22 Lu XL,Bardet J-P,Huang MS.Numerical solutions of strain localization with nonlocal softening plasticity.Computer Methods in Applied Mechanics and Engineering,2009,198(47):3702-3711
23 Lu XL,Bardet J-P,Huang MS.Spectral analysis of nonlocal regularization in two-dimensional finite element models.InternationalJournal of Numerical and Analytical Methods in Geomechanics,2012,36(2):219-235
24 Bazant ZP,Lin F-B.Nonlocal smeared cracking model for concrete fracture.Journal of structural Engineering,1988,114(11):2493-2510
25 Eringen AC.Nonlocal Continuum Field Theories.New York:Springer-Verlag,2002
26 Nilsson C.Nonlocal strain softening bar revisited.International Journal of Solids and Structures,1997,34(34):4399-4419
27 Borino G,Fuschi P,Polizzotto C.A thermodynamic approach to nonlocal plasticity and related variational principles.Journal of Applied Mechanics,1999,66(4):952-963
28 de Sciarra FM.A general theory for nonlocal softening plasticity of integral-type.International Journal of Plasticity,2008,24(8):1411-1439
29 武守信,魏吉瑞,杨舒蔚.基于能量等效原理的应变局部化分析I:一维解析解.力学学报,2017,49(3):(Wu Shouxin,We I Jirui,Yang Shuwei.Analysis of strain localization by energy equivalence between local and nonlocal state spaces-part I:one-dimensional analytical solution.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):(in Chinese))
30 冯春,李世海,刘晓宇.基于颗粒离散元法的连接键应变软化模型及其应用.力学学报,2016,48(1):76-85(Feng Chun,Li Shihai,Liu Xiaoyu.Particle-dem based linked bar strain softening model and its application.Chinese Journal of Theoretical and Applied Mechanics,2016,48(1):76-85(in Chinese))
31 万征,姚仰平,孟达.复杂加载下混凝土的弹塑性本构模型.力学学报,2016,48(5):1159-1171(Wan Zheng,Yao Yangping,Meng Da.An elastoplastic constitutive model of concrete under complicated load.Chinese Journal of Theoretical and Applied Mechanics,2016,48(5):1159-1171(in Chinese))
32 Simo JC,Hughes TJR.Computational Inelasticity.New York:Springer-Verlag,1998
33 Regueiro RA,Borja RI.A finite element model of localized deformation in frictional materials taking a strong discontinuity approach.Finite Elements in Analysis and Design,1999,33:283-315
34 Yumlu M,Ozbay MU.A study of the behavior of brittle rocks under plan strain and triaxial loading condition.International Journal of Rock Mechanics and Mining Science&Geomechanics Abstracts,1995,32(7):725-733
35 韩铁林,师俊平,陈蕴生等.轴、侧向同卸荷下砂岩力学特性影响的试验研究.力学学报,2016,48(3):936-943(Han Tielin,Shi Junping,Chen Yunsheng,et al.Experimental study on mechanics characteristics of sandstone under axial unloading and radial unloading path.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):936-943(in Chinese))
2 钱建固,吕玺琳,黄茂松.平面应变状态下土体的软化特性与本构模拟.岩土力学,2009,30(3):617-622(Qian Jiangu,L¨u Xilin,Huang Maosong.Softening characteristics of soils and constitutive modeling under plane strain condition.Rock and Soil Mechanics,2009,30(3):617-622(in Chinese))
3 徐连民,王天竹,祁德庆等.岩土中的剪切带局部化问题研究:回顾与展望.力学季刊,2004,25(4):484-489(Xu Lianmin,Wang Tianzhu,Qi Deqing,et al.Study on geotechnical shear band localization:retrospect and prospect.Chinese Quarterly of Mechanics,2004,25(4):484-489(in Chinese))
4 张启辉,赵锡宏.粘性土局部化剪切带变形的机理研究.岩土力学,2002,23(1):31-35(Zhang Qihui,Zhao Xihong.Study on mechanism of localized shear band formation in clay.Rock and Soil Mechanics,2002,23(1):31-35(in Chinese))
5 王杰,李世海,张青波.基于单元破裂的岩石裂纹扩展模拟方法.力学学报,2015,47(1):105-118(Wang Jie,Li Sihai,Zhang Qingbo.Simulation of crack propagation of rock based on splitting elements.Chinese Journal of Theoretical and Applied Mechanics,2015,47(1):105 -118(in Chinese))
6 李宏儒,胡再强,冯飞等.结构性黄土二元介质本构模型在局部化剪切带中的应用.岩土力学,2012,33(9):2803-2844(Li Hongru,Hu Zaiqiang,Feng Fei,et al.Application of structural loess binarymedium model to localization shear band.Rock and Soil Mechanics,2012,33(9):2803-2844(in Chinese))
7 黄茂松,贾苍琴,钱建固.岩土材料应变局部化的有限元分析方法.计算力学学报,2007,24(4):465-471(Huang Maosong,Jia Cangqin,Qian Jiangu.Strain localization problems in geomaterials using finite elements.Chinese Journal of Computational Mechanics,2007,24(4):465-471(in Chinese))
8 邵龙潭,刘港,郭晓霞.三轴试样破坏后应变局部化影响的实验研究.岩土工程学报,2016,38(3):385-394(Shao Longtan,Liu Gang,Guo Xiaoxia.Effects of strain localization of triaxial samples in post-failure state.Chinese Journal of Rock Mechanics and Engineering,2016,38(3):385-394(in Chinese))
9 李学丰,黄茂松,钱建固.基于非共轴理论的各向异性砂土应变局部化分析.工程力学,2014,31(3):205-246(Li Xuefeng,Huang Maosong,Qian Jiangu.Strain localization analysis of anisotropic sands based on non-coaxial theory.Engineering Mechanics,2014,31 (3):205-246(in Chinese))
10 王水林,郑宏,刘泉声等.应变软化岩体分析原理及其应用,岩土力学,2014,35(3):609-630(Wang Shuilin,Zheng Hong,Liu Quansheng,et al.Principle of analysis of strain-softening rock mass and its application.Rock and Soil Mechanics,2014,35(3):609-630(in Chinese))
11 Read HE,Hegemier GA.Strain softening of rock,soil and concrete—a review article.Mechanics and Materials,1984,3(3):271-294
12 Vardoulakis I.Shear band inclination and shear modulus of sand in biaxial test.International Journal for Numerical and Analytical Methods in Geomechanics,1980,4(2):103-119
13 Jir′asek M,Ba?zant ZP.Inelastic analysis of structures.Chichester,England:John Wiley&Sons,2002
14 Pietruszczak ST,Mr′oz Z.Finite element analysis of deformation of strain-softening materials.International Journal for Numerical Methods in Engineering,1981,17(3):327-334
15 Ortiz M,Leroy Y,Needleman A.A finite element method for localized failure analysis.Computer Methods in Applied Mechanics and Engineering,1987,61(2):189-214
16 de Borst R,Sluys LJ,M¨uhlhaus H-B,et al.Fundamental issues in finite element analyses of localization of deformation.Engineering Computations,1993,10(2):99-121
17 Simo JC.Strain softening and dissipation:a unification of approaches//Mazars J,Ba?zant JP,eds.Cracking and Damage:strain localization and size effect.London and New York:Elsevier Applied Science,1988:440-461
18 de Borst R,M¨uhlhaus H-B.Gradient-dependent plasticity:formulation and algorithm aspects.International Journal for Numerical Methods in Engineering,1992,35(3):521-539
19 Luzio GD,Ba?zant ZP.Spectral analysis of localization in nonlocal and over-nonlocal materials with softening plasticity or damage.International Journal of Solids and Structures,2005,42(23):6071-6100
20 Vermeer PA,Brinkgreve RBJ.A new effective non-local strain measure for softening plasticity//Chambon V,Desrues J,Vardoulakis I,eds.Localisation and Bifurcation Theory for Soils and Rocks.Rotterdam,Netherland:A.A.Balkema,1994
21 Stromberg L,Ristinmaa M.FE-formulation of a nonlocal plasticity theory.Computer Methods in Applied Mechanics and Engineering,1996,136(1-2):127-144
22 Lu XL,Bardet J-P,Huang MS.Numerical solutions of strain localization with nonlocal softening plasticity.Computer Methods in Applied Mechanics and Engineering,2009,198(47):3702-3711
23 Lu XL,Bardet J-P,Huang MS.Spectral analysis of nonlocal regularization in two-dimensional finite element models.InternationalJournal of Numerical and Analytical Methods in Geomechanics,2012,36(2):219-235
24 Bazant ZP,Lin F-B.Nonlocal smeared cracking model for concrete fracture.Journal of structural Engineering,1988,114(11):2493-2510
25 Eringen AC.Nonlocal Continuum Field Theories.New York:Springer-Verlag,2002
26 Nilsson C.Nonlocal strain softening bar revisited.International Journal of Solids and Structures,1997,34(34):4399-4419
27 Borino G,Fuschi P,Polizzotto C.A thermodynamic approach to nonlocal plasticity and related variational principles.Journal of Applied Mechanics,1999,66(4):952-963
28 de Sciarra FM.A general theory for nonlocal softening plasticity of integral-type.International Journal of Plasticity,2008,24(8):1411-1439
29 武守信,魏吉瑞,杨舒蔚.基于能量等效原理的应变局部化分析I:一维解析解.力学学报,2017,49(3):(Wu Shouxin,We I Jirui,Yang Shuwei.Analysis of strain localization by energy equivalence between local and nonlocal state spaces-part I:one-dimensional analytical solution.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):(in Chinese))
30 冯春,李世海,刘晓宇.基于颗粒离散元法的连接键应变软化模型及其应用.力学学报,2016,48(1):76-85(Feng Chun,Li Shihai,Liu Xiaoyu.Particle-dem based linked bar strain softening model and its application.Chinese Journal of Theoretical and Applied Mechanics,2016,48(1):76-85(in Chinese))
31 万征,姚仰平,孟达.复杂加载下混凝土的弹塑性本构模型.力学学报,2016,48(5):1159-1171(Wan Zheng,Yao Yangping,Meng Da.An elastoplastic constitutive model of concrete under complicated load.Chinese Journal of Theoretical and Applied Mechanics,2016,48(5):1159-1171(in Chinese))
32 Simo JC,Hughes TJR.Computational Inelasticity.New York:Springer-Verlag,1998
33 Regueiro RA,Borja RI.A finite element model of localized deformation in frictional materials taking a strong discontinuity approach.Finite Elements in Analysis and Design,1999,33:283-315
34 Yumlu M,Ozbay MU.A study of the behavior of brittle rocks under plan strain and triaxial loading condition.International Journal of Rock Mechanics and Mining Science&Geomechanics Abstracts,1995,32(7):725-733
35 韩铁林,师俊平,陈蕴生等.轴、侧向同卸荷下砂岩力学特性影响的试验研究.力学学报,2016,48(3):936-943(Han Tielin,Shi Junping,Chen Yunsheng,et al.Experimental study on mechanics characteristics of sandstone under axial unloading and radial unloading path.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):936-943(in Chinese))