ZHAO Bing, LIU Tao, HE Jian-hui, ZHU Hao-rui, LI Wei. A Modified Gradient Elastic Theory Considering Damage[J]. Applied Mathematics and Mechanics, 2017, 38(9): 999-1008. doi: 10.21656/1000-0887.370285
Citation: ZHAO Bing, LIU Tao, HE Jian-hui, ZHU Hao-rui, LI Wei. A Modified Gradient Elastic Theory Considering Damage[J]. Applied Mathematics and Mechanics, 2017, 38(9): 999-1008. doi: 10.21656/1000-0887.370285

A Modified Gradient Elastic Theory Considering Damage

doi: 10.21656/1000-0887.370285
  • Received Date: 2016-09-20
  • Rev Recd Date: 2016-11-13
  • Publish Date: 2017-09-15
  • To describe the material mechanics behaviors depending on microstructure, the gradient elastic theory with significant advantages was investigated. The gradient elastic theory was combined with the damage theory to consider the influence of microstructure on material failure. Then a modified gradient elasticity damage theory was proposed, based on which the basic law of thermodynamics, the strain tensor, the damage variable and the scalar strain gradient tensor were taken as the state variables of the Helmholtz free energy. The Taylor expansion of the Helmholtz free energy function was conducted near the natural state, and the general expressions of the modified gradient elasticity damage constitutive functions were derived. The finite element code was programmed to simulate the development process of damage localization in soil specimens. The results show that, the traditional mesh dependence in numerical simulation can be removed under the modified gradient elasticity damage theory. The band of the damage localization does not concur with the damage, but occurs after the damage development to some extent.
  • loading
  • [1]
    Askes H, Aifantis E C. Numerical modeling of size effects with gradient elasticity—formulation, meshless discretization and examples[J]. International Journal of Fracture,2002,117(4): 347-358.
    SONG Zhan-ping, ZHAO Bing, HE Jian-hui, et al. Modified gradient elasticity and its finite element method for shear boundary layer analysis[J]. Mechanics Research Communications,2014,62: 146-154.
    Cosserat E, Cosserat F. Théorie des corps déformables[J]. Hermann Archives,1909.
    Toupin R A. Elastic materials with couple-stresses[J]. Archive for Rational Mechanics and Analysis,1962,11(1): 385-414.
    Toupin R A. Theories of elasticity with couple-stress[J]. Archive for Rational Mechanics and Analysis,1964,17(2): 85-112.
    Mindlin R D, Tiersten H F. Effects of couple-stresses in linear elasticity[J]. Archive for Rational Mechanics and Analysis,1962,11(1): 415-448.
    Mindlin R D. Micro-structure in linear elasticity[J]. Archive for Rational Mechanics and Analysis,1964,16(1): 51-78.
    Mindlin R D. Second gradient of strain and surface-tension in linear elasticity[J]. International Journal of Solids and Structures,1965,1(4): 417-438.
    Mindlin R D. Theories of Elastic Continua and Crystal Lattice Theories [M]//Krner E, ed. Mechanics of Generalized Continua . Berlin: Springer, 1968: 312-320.
    Eringen A C. Theories of nonlocal plasticity[J]. International Journal of Engineering Science,1983,21(7): 741-751.
    Eringen A C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves[J]. Journal of Applied Physics,1983,54(9): 4703-4710.
    Triantafyllidis N, Aifantis E C. A gradient approach to localization of deformation: I. hyperelastic materials[J]. Journal of Elasticity,1986,16(3): 225-237.
    Aifantis E C. On the role of gradients in the localization of deformation and fracture[J]. International Journal of Engineering Science,1992,30(10): 1279-1299.
    Altan S B, Aifantis E C. On the structure of the mode III crack-tip in gradient elasticity[J]. Scripta Metallurgica et Materialia,1992,26(2): 319-324.
    Ru C Q, Aifantis E C. A simple approach to solve boundary-value problems in gradient elasticity[J]. Acta Mechanica,1993,101(1): 59-68.
    徐晓建, 邓子辰. 多层简化应变梯度Timoshenko梁的变分原理分析[J]. 应用数学和力学, 2016,37(3): 235-244. (XU Xiao-jian, DENG Zi-chen. The variational principle for multi-layer Timoshenko beam systems based on the simplified strain gradient theory[J]. Applied Mathematics and Mechanics,2016,37(3): 235-244.(in Chinese))
    Pijaudier-Cabot G, Baant Z P. Nonlocal damage theory[J]. Journal of Engineering Mechanics,1987,113(10): 1512-1533.
    Baant Z P, Jirásek M. Nonlocal integral formulations of plasticity and damage: survey of progress[J]. Journal of Engineering Mechanics,2002,128(11): 1119-1149.
    Baant Z P, Pijaudier-Cabot G. Nonlocal continuum damage, localization instability and convergence[J]. Journal of Applied Mechanics,1988,55(2): 287-293.
    Frémond M, Nedjar B. Damage, gradient of damage and principle of virtual power[J]. International Journal of Solids and Structures,1996,33(8): 1083-1103.
    赵吉东, 周维垣, 刘元高, 等. 岩石类材料应变梯度损伤模型研究及应用[J]. 水利学报, 2002(7): 70-74.(ZHAO Ji-dong, ZHOU Wei-yuan, LIU Yuan-gao, et al. Strain gradient enhanced damage model for rock material and its application[J]. Journal of Hydraulic Engineering,2002(7): 70-74.(in Chinese))
    唐雪松, 蒋持平, 郑健龙. 各向同性弹性损伤本构方程的一般形式[J]. 应用数学和力学, 2001,22(12): 1317-1323.(TANG Xue-song, JIANG Chi-ping, ZHENG Jian-long. General expressions of constitutive equations for isotropic elastic damaged materials[J]. Applied Mathematics and Mechanics,2001,22(12): 1317-1323.(in Chinese))
    沈珠江. 结构性粘土的非线性损伤力学模型[J]. 水利水运科学研究, 1993(3): 247-255.(SHEN Zhu-jiang. A nonlinear damage model for structured clay[J]. Hydro-Science and Engineering,1993(3): 247-255.(in Chinese))
    Belytschko T, Baant Z P, Yul-Woong H, et al. Strain-softening materials and finite-element solutions[J]. Computers & Structures, 1986,23(2): 163-180.
    Baant Z P, CHANG Ta-peng. Nonlocal finite element analysis of strain-softening solids[J]. Journal of Engineering Mechanics,1987,113(1): 89-105.
  • 加载中


    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (647) PDF downloads(943) Cited by()
    Proportional views


    DownLoad:  Full-Size Img  PowerPoint