First-order gradient damage theory

• Published in: Applied mathematics and mechanics Vol. 31; no. 8; pp. 987 - 994
• Main Author: 赵冰 郑颖人 曾明华 唐雪松 李小纲
• Format: Journal Article
• Language: English
• Published: Heidelberg Shanghai University Press 01.08.2010; School of Civil Engineering and Architecture, Changsha University of Science and Technology, Changsha 410114, P. R. C%Department of Architectural Engineering, Logistical Engineering University of the People's Liberation Army, Chongqing 400041, P. R. China%School of Civil Engineering and Architecture, Changsha University of Science and Technology, Changsha 410114, P. R. China%School of Mathematics and Computer Science, Ningxia University,Yinchuan 750021, P. R. China; Department of Architectural Engineering, Logistical Engineering University of the People's Liberation Army, Chongqing 400041, P. R. China
• Subjects: Applications of Mathematics; Classical Mechanics; Fluid- and Aerodynamics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Helmholtz free energy; 74R05; thermodynamics; O346.5; damage gradient; damage localization; constitutive functional expansion method
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-010-1334-9

Taking the strain tensor, the scalar damage variable, and the damage gradient as the state variables of the Helmholtz free energy, the general expressions of the firstorder gradient damage constitutive equations are derived directly from the basic law of irreversible thermodynamics with the constitutive functional expansion method at the natural state. When the damage variable is equal to zero, the expressions can be simplified to the linear elastic constitutive equations. When the damage gradient vanishes, the expressions can be simplified to the classical damage constitutive equations based on the strain equivalence hypothesis. A one-dimensional problem is presented to indicate that the damage field changes from the non-periodic solutions to the spatial periodic-like solutions with stress increment. The peak value region develops a localization band. The onset mechanism of strain localization is proposed. Damage localization emerges after damage occurs for a short time. The width of the localization band is proportional to the internal characteristic length.