ZHAO Bing, ZHENG Ying-ren, ZENG Ming-hua, TANG Xue-song, LI Xiao-gang. First-Order Gradient Damage Theory[J]. Applied Mathematics and Mechanics, 2010, 31(8): 941-948. doi: 10.3879/j.issn.1000-0887.2010.08.007
 Citation: ZHAO Bing, ZHENG Ying-ren, ZENG Ming-hua, TANG Xue-song, LI Xiao-gang. First-Order Gradient Damage Theory[J]. Applied Mathematics and Mechanics, 2010, 31(8): 941-948.

##### doi: 10.3879/j.issn.1000-0887.2010.08.007
• Rev Recd Date: 2010-06-21
• Publish Date: 2010-08-15
• Taking the strain tensor, scalar damage variable and damage gradient as the state variables of Helmholtz free energy, the general expressions of first-order gradient damage constitutive equations were derived directly from the basic law of irreversible thermodynamics by constitutive functional expansion method at natural state. When damage variable was equal to zero, the expressions could be simplified to linear elastic constitutive equations; when the damage gradient vanished, the expressions could be smiplified to the classical damage constitutive equations based on the strain equivalence hypothesis. One-dimensional problem is presented to indicate that the damage field changes from non-periodic solutions to the spatial periodic-like solutions with stress in crement. The peak value region developes to a localization band. The onsetm echanism of strain localization is advised. Damage localization emerges after damage occurs for a short tmie. The width of localization band is proportional to the internal characteristic length.
•  [1] 李锡夔, 刘泽佳, 严颖.饱和多孔介质中的混合有限元法和有限应变下应变局部化分析[J]. 力学学报, 2003, 35(6):668-676. [2] 张洪武.应变局部化分析中两类不同材料模型的讨论[J].力学学报, 2003, 35(1):80-84. [3] 赵冰, 李宁, 盛国刚.软化岩土介质的应变局部化研究进展——意义·现状·应变梯度[J].岩土力学, 2005, 26(3):111-118. [4] Baant Z P, Chang Ta-peng. Non-local finite element analysis of strain-softening solids [J]. J Eng Mech-ASCE, 1987, 113(1): 89-105. [5] Belytschko T. Strain-softening materials and finite-element solutions[J]. Int J Solids Structures, 1986, 23(2): 163-180. [6] 陈少华, 王自强.应变梯度理论进展[J].力学进展, 2003, 33(2): 207-216. [7] Eringen A C. Nonlocal polar elastic continua [J]. Int J Solids Structures, 1972, 10(1): 233- 248. [8] de Borst R, Mühlhaus H B. Gradient-dependent plasticity: formulation and algorithmic aspects [J]. Int J Numer Meth Eng, 1992, 35(3):521-539. [9] Fleck N A, Hutchinson J W. A phenomenological theory for strain gradient effects in plasticity[J]. J Mech Phys Solids, 1993, 41(12):1825-1857. [10] Ellen K, Ekkehard R, de Borst R. An anisotropic gradient damage model for quasi-brittle materials [J]. Compu Methods Appl Mech Eng, 2000, 183(1): 87-103. [11] Pijaudier G, Baant Z P. Nonlocal damage theory[J]. J Eng Mech-ASCE, 1987, 113(10): 1512-1533. [12] Baant Z P, LIN Feng-bao. Nonlocal smeared cracking model for concrete fracture [J]. Journal of Structural Engineering, 1988, 114(11): 2493-2510. [13] Baant Z P, Pijaudier G. Nonlocal continuum damage, localization Instability and conver- gence [J]. J Appl Mech-ASCE, 1988, 55(2):287-293. [14] Baant Z P. Non-local damage theory based on micro-mechanics of crack interactions [J]. J Eng Mech-ASCE, 1994, 120(3): 593-617. [15] Fremond M, Nedjar B. Damage, gradient of damage and principle of virtual power[J]. Int J Solids Structures, 1996, 33(8): 1083-1103. [16] Wei Y, Hutchinson J W. Steady-state crack growth and work of fracture for solids characterized by strain gradient plasticity[J].J Mech Phys Solids, 1997, 45(8): 1253-1273. [17] Chen S, Wang T. A new hardening law for strain gradient plasticity[J]. Acta Mater, 2000, 48(16): 3997-4005. [18] 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. [19] 赵启林.力学仿真中应变软化问题数模分析与神经计算力学研究[D].博士学位论文.南京:河海大学, 2001: 42-43. [20] 孙庆平, 赵智军, 卿新林, 陈渭泽, 戴福隆. ZrO2相变多晶体塑性变形局部化行为的宏观-细观实验研究[J].中国科学(A辑), 1994, 24(4):383-388.

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