ANALYSIS OF STRAIN LOCALIZATION BY ENERGY EQUIVALENCE: Ⅰ. ONE-DIMENSIONAL ANALYTICAL SOLUTION
-
摘要: 基于热力学第一定律和非局部塑性理论，提出了一种求解应变局部化问题的非局部方法。对材料的每一点定义了局部和非局部两种状态空间，局部状态空间的内变量通过非局部权函数映射到非局部空间，成为非局部内变量。在应变软化过程中，局部状态空间中的塑性变形服从正交流动法则，材料的软化律在非局部状态空间中被引入。通过两个状态空间的塑性应变能耗散率的等效，得到了应变软化过程中明确定义的局部化区域以及其中的塑性应变分布。应用本方法导出了一维应变局部化问题的解析解。解析解表明，应变局部化区域的尺寸只与材料内尺度有关；对于高斯型非局部权函数，局部化区域的尺寸大约是材料内尺度的6倍。一维算例表明，局部化区域的塑性应变分布以及载荷-位移曲线仅与材料参数和结构几何尺寸有关，变形局部化区域的尺寸随着材料内尺度的减小而减小，同时塑性应变也随着材料内尺度的减小变得更加集中。当内尺度趋近于零时，应用本文方法得到的解与采用传统的局部塑性理论得到的解相同。Abstract: Based on the first law of thermodynamics and the nonlocal plasticity theories, a new approach is proposed to solve the strain localization problems induced by strain softening. For each material point, two state spaces, local and nonlocal state spaces, are defined such that the local internal variable can be mapped, from the local state space by integral transformation with the nonlocal weighting function, into the nonlocal internal variable in the nonlocal state space. During strain softening, the plastic deformation follows the normal flow rule in the local state space and the softening law is introduced in the nonlocal state space. It is assumed that the strain softening is a global material behavior and the plastic energy dissipation within the entire material body is always positive. However, the balance of momentum is still satisfied locally. By equating the rates of the plastic energy dissipation in the two state spaces during strain softening, the localization zone and the plastic strain distribution become well-defined. Analytical solution for the one-dimensional strain localization is developed, and it is well shown that the plastic strain distribution and load-displacement curves are well-defined by the material properties, such as the softening modulus and internal length scale, as well as the geometry of the material body. For the Gaussian-type weighting functions the width of the localization zone is approximately six times the internal length scale. Numerical example demonstrates that the size of the localization zone decreases as the internal length scale is reduced, and the distribution of the plastic strain in the localization zone becomes more concentrated when the internal length scale becomes smaller. As the internal length scale approaches to zero, the solution reduces to the one predicted by the conventional local plasticity theory.
-
Key words:
- strain localization /
- nonlocal plasticity /
- internal length scale /
- mesh-dependence /
- finite element
-
图 2 不同材料内尺度下拉杆内的塑性应变分布
Figure 2. Plastic strain distributions along the axis of the bar for different internal length scales
-
[1] Read HE, Hegemier GA. Strain softening of rock, soil and concrete-a review article. Mechanics and Materials, 1984, 3(3):271-294 http://www.sciencedirect.com/science/article/pii/0167663684900280 [2] 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 doi: 10.1002/(ISSN)1096-9853 [3] Jirásek M, Bažant ZP. Inelastic Analysis of Structures. Chichester, England:John Wiley & Sons, 2002 [4] 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 http://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201603002.htmShao 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) http://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201603002.htm [5] 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 http://www.cnki.com.cn/Article/CJFDTOTAL-GCLX201403029.htmLi 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) http://www.cnki.com.cn/Article/CJFDTOTAL-GCLX201403029.htm [6] 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 http://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201403001.htmWang 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) http://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201403001.htm [7] 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 http://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200903009.htmQian Jiangu, Lü 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) http://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200903009.htm [8] 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 doi: 10.6052/0459-1879-14-239Wang Jie, Li Shihai, 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) doi: 10.6052/0459-1879-14-239 [9] Pietruszczak ST, Mróz Z. Finite element analysis of deformation of strain-softening materials. International Journal for Numerical Methods in Engineering, 1981, 17(3):327-334 doi: 10.1002/(ISSN)1097-0207 [10] 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 doi: 10.1016/0045-7825(87)90004-1 [11] de Borst R, Sluys LJ, Mühlhaus HB, et al. Fundamental issues in finite element analyses of localization of deformation. Engineering Computations, 1993, 10(2):99-121 doi: 10.1108/eb023897 [12] Simo JC. Strain softening and dissipation:a unification of approaches//Mazars J, Bažant ZP, eds. Cracking and Damage:Strain Localization and Size Effect. London and New York:Elsevier Applied Science, 1988:440-461 [13] Schaeffer DG. Instability and ill-posedness in the deformation of granular materials. International Journal of Numerical and Analytical Methods in Geomechanics, 1990, 14(4):253-278 doi: 10.1002/(ISSN)1096-9853 [14] de Borst R, Mühlhaus HB. Gradient-dependent plasticity:formulation and algorithm aspects. International Journal for Numerical Methods in Engineering, 1992, 35(3):521-539 doi: 10.1002/(ISSN)1097-0207 [15] 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 [16] Stromberg L, Ristinmaa M. FE-formulation of a nonlocal plasticity theory. Computer Methods in Applied Mechanics and Engineering, 1996, 136(1-2):127-144 doi: 10.1016/0045-7825(96)00997-8 [17] Lü XL, Bardet JP, Huang MS. Numerical solutions of strain localization with nonlocal softening plasticity. Computer Methods in Applied Mechanics and Engineering, 2009, 198(47):3702-3711 (in Chinese) http://www.sciencedirect.com/science/article/pii/S0045782509002552 [18] Lü XL, Bardet JP, Huang MS. Spectral analysis of nonlocal regularization in two-dimensional finite element models. International Journal of Numerical and Analytical Methods in Geomechanics, 2012, 36(2):219-235 doi: 10.1002/nag.v36.2 [19] Eringen AC. Nonlocal Continuum Field Theories. New York:Springer-Verlag, 2002 [20] Bažant ZP, Lin FB. Nonlocal smeared cracking model for concrete fracture. Journal of structural Engineering, 1988, 114(11):2493-2510 doi: 10.1061/(ASCE)0733-9445(1988)114:11(2493) [21] Luzio GD, Bažant 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 doi: 10.1016/j.ijsolstr.2005.03.038 [22] Jirásek M, Rolshoven S. Comparison of integral-type nonlocal plasticity models for strain-softening materials. International Journal of Engineering Science, 2003, 41(13-14):1553-1602 doi: 10.1016/S0020-7225(03)00027-2 [23] Nilsson C. Nonlocal strain softening bar revisited. International Journal of Solids and Structures, 1997, 34(34):4399-4419 http://www.sciencedirect.com/science/article/pii/S002076839700019X [24] 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 doi: 10.1115/1.2791804 [25] de Sciarra FM. A general theory for nonlocal softening plasticity of integral-type. International Journal of Plasticity, 2008, 24(8):1411-1439 doi: 10.1016/j.ijplas.2007.09.011 [26] Nguyen GD. A thermodynamic approach to non-local damage modeling of concrete. International Journal of Solids and Structures, 2008, 45(7-8):1918-1934 doi: 10.1016/j.ijsolstr.2007.11.001 [27] Ogata K. Modern Control Engineering. 5th Edition. New Jersey:Prentice Hall, 2010 [28] Tarn JQ. A state space formalism for anisotropic elasticity. Part I:Rectilinear anisotropy. International Journal of Solids and Structures, 2002, 39(20):5157-5172 doi: 10.1016/S0020-7683(02)00412-2 [29] Simo JC, Hughes TJR. Computational Inelasticity. New York:Springer-Verlag, 1998 [30] Reddy JN. An Introduction to Continuum Mechanics. New York:Cambridge University Press, 2008 [31] 陈昀, 金衍, 陈勉.基于能量耗散的岩石脆性评价方法.力学学报, 2015, 47(6):984-993 doi: 10.6052/0459-1879-15-156Chen Yun, Jin Yan, Chen Mian. A rock brittleness evaluation method based on energy dissipation. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(6):984-993 (in Chinese doi: 10.6052/0459-1879-15-156 [32] Bažant Z, Pijaudier-Cabot G. Measurement of characteristic length of nonlocal continuum. Journal of Engineering Mechanics, 1989, 115(4):755-767 doi: 10.1061/(ASCE)0733-9399(1989)115:4(755) [33] Mühlhaus H-B, Vardoulakis I. The thickness of shear bands in granular materials. Géotechnique, 1987, 37(3):271-283 doi: 10.1680/geot.1987.37.3.271 [34] Lü XL, Bardet JP, Huang MS. Length scales interaction in nonlocal plastic strain localization of bars of varying section. Journal of Engineering Mechanics, 2010, 136(8):1036-1042 doi: 10.1061/(ASCE)EM.1943-7889.0000145 -
下载:
点击查看大图
图(3)
计量
- 文章访问数: 1007
- HTML全文浏览量: 163
- PDF下载量: 550
- 被引次数: 0
