考虑损伤的修正梯度弹性理论
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摘要
梯度弹性理论在描述材料微结构起主导作用的力学行为时具有显著优势,将其与损伤理论相结合,可在材料破坏研究中考虑微结构的影响.基于修正梯度弹性理论,将应变张量、应变梯度张量和损伤变量作为Helmholtz自由能函数的状态变量,并在自然状态附近对自由能函数作Taylor展开,进而由热力学基本定律,推导出修正梯度弹性损伤理论本构方程的一般形式.编制有限元程序,模拟土样损伤局部化带的发展演化过程.结果表明,修正梯度弹性损伤理论消除了网格依赖性;损伤局部化带不是与损伤同时发生,而是在损伤发展到一定程度后再逐渐显现出来.
To describe the material mechanics behaviors depending on microstructure,the gradient 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.
To describe the material mechanics behaviors depending on microstructure,the gradient 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.
引文
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[23]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 C hinese))
[24]Belytschko T,Ba6ant Z P,Yul-W oong H,et al.Strain-softening materials and finite-element solutions[J].Computers&Structures,1986,23(2):163-180.
[25]Ba6ant Z P,CHANG T a-peng.Nonlocal finite element analysis of strain-softening solids[J].Journal of Engineering M echanics,1987,113(1):89-105.
[2]SONG Zhan-ping,ZHAO Bing,HE Jian-hui,et al.M odified gradient elasticity and its finite elem ent m ethod for shear boundary layer analysis[J].M echanics Research Communications,2014,62:146-154.
[3]Cosserat E,Cosserat F.T héorie des corps déformables[J].Hermann Archives,1909.
[4]T oupin R A.Elastic materials with couple-stresses[J].Archive for Rational Mechanics and Analysis,1962,11(1):385-414.
[5]T oupin R A.T heories of elasticity with couple-stress[J].Archive for Rational Mechanics and Analysis,1964,17(2):85-112.
[6]M indlin R D,T iersten H F.Effects of couple-stresses in linear elasticity[J].Archive for Rational Mechanics and A nalysis,1962,11(1):415-448.
[7]M indlin R D.M icro-structure in linear elasticity[J].Archive for Rational Mechanics and Analysis,1964,16(1):51-78.
[8]M indlin R D.Second gradient of strain and surface-tension in linear elasticity[J].International Journal of Solids and Structures,1965,1(4):417-438.
[9]M indlin R D.Theories of Elastic Continua and Crystal Lattice Theories[M]//Kr9ner E,ed.Mechanics of G eneralized Continua.Berlin:Springer,1968:312-320.
[10]Eringen A C.T heories of nonlocal plasticity[J].International Journal of Engineering Science,1983,21(7):741-751.
[11]Eringen A C.On differential equations of nonlocal elasticity and solutions of screw dislocation and surface w aves[J].J ournal of A pplied Physics,1983,54(9):4703-4710.
[12]T riantafyllidis N,Aifantis E C.A gradient approach to localization of deformation:I.hyperelastic m aterials[J].J ournal of Elasticity,1986,16(3):225-237.
[13]Aifantis E C.On the role of gradients in the localization of deformation and fracture[J].International J ournal of Engineering Science,1992,30(10):1279-1299.
[14]Altan S B,Aifantis E C.On the structure of the mode III crack-tip in gradient elasticity[J].Scripta M etallurgica et M aterialia,1992,26(2):319-324.
[15]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.
[16]徐晓建,邓子辰.多层简化应变梯度T imoshenko梁的变分原理分析[J].应用数学和力学,2016,37(3):235-244.(X U X iao-jian,D EN G Z i-chen.T he variational principle for m ulti-layer T im oshenko beam system s based on the sim plified strain gradient theory[J].A pplied M athematics and M echanics,2016,37(3):235-244.(in C hinese))
[17]Pijaudier-Cabot G,Ba6ant Z P.Nonlocal damage theory[J].Journal of Engineering Mechanics,1987,113(10):1512-1533.
[18]Ba6ant Z P,Jirásek M.Nonlocal integral formulations of plasticity and damage:survey of progress[J].Journal of Engineering Mechanics,2002,128(11):1119-1149.
[19]Ba6ant Z P,Pijaudier-Cabot G.Nonlocal continuum damage,localization instability and convergence[J].J ournal of A pplied M echanics,1988,55(2):287-293.
[20]Frémond M,Nedjar B.Damage,gradient of damage and principle of virtual power[J].International J ournal of Solids and Structures,1996,33(8):1083-1103.
[21]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 W ei-yuan,LIU Yuan-gao,et al.Strain gradient enhanced dam age m odel for rock m aterial and its application[J].J ournal of H ydraulic Engineering,2002(7):70-74.(in C hinese))
[22]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.(T AN G X ue-song,JIAN G C hi-ping,Z HEN G Jian-long.G eneral expressions of constitutive equations for isotropic elastic dam aged m aterials[J].A pplied M athematics and M echanics,2001,22(12):1317-1323.(in C hinese))
[23]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 C hinese))
[24]Belytschko T,Ba6ant Z P,Yul-W oong H,et al.Strain-softening materials and finite-element solutions[J].Computers&Structures,1986,23(2):163-180.
[25]Ba6ant Z P,CHANG T a-peng.Nonlocal finite element analysis of strain-softening solids[J].Journal of Engineering M echanics,1987,113(1):89-105.