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 d...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.展开更多
This paper addresses the modeling of fracture in quasi-brittle materials using a phase-field approach to the description of crack topol-ogy.Within the computational mechanics community,several studies have treated the...This paper addresses the modeling of fracture in quasi-brittle materials using a phase-field approach to the description of crack topol-ogy.Within the computational mechanics community,several studies have treated the issue of modeling fracture using phase fields.Most of these studies have used an approach that implies the lack of a damage threshold.We herein explore an alternative model that includes a damage threshold and study how it compares with the most popular approach.The formulation is systematically explained within a rigorous variational framework.Subsequently,we present the corresponding three-dimensional finite element discretization that leads to a straightforward numerical implementation.Benchmark simulations in two dimensions and three dimensions are then presented.The results show that while an elastic stage and a damage threshold are ensured by the present model,good agreement with the results reported in the literature can be obtained,where such features are generally absent.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 50978036)the Natural Science Foundation of Hunan Province of China (No. 09JJ6080)the Applied Basic Research Programs of Ministry of Transportation of China (No. 2009-319-825-100)
文摘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.
文摘This paper addresses the modeling of fracture in quasi-brittle materials using a phase-field approach to the description of crack topol-ogy.Within the computational mechanics community,several studies have treated the issue of modeling fracture using phase fields.Most of these studies have used an approach that implies the lack of a damage threshold.We herein explore an alternative model that includes a damage threshold and study how it compares with the most popular approach.The formulation is systematically explained within a rigorous variational framework.Subsequently,we present the corresponding three-dimensional finite element discretization that leads to a straightforward numerical implementation.Benchmark simulations in two dimensions and three dimensions are then presented.The results show that while an elastic stage and a damage threshold are ensured by the present model,good agreement with the results reported in the literature can be obtained,where such features are generally absent.
