Damaged plasticity model for concrete and other quasi-brittle materialsProducts: Abaqus/Standard Abaqus/Explicit This section describes the concrete damaged plasticity model provided in Abaqus for the analysis of concrete and other quasi-brittle materials. The material library in Abaqus also includes other constitutive models for concrete based on the smeared crack approach. These are the smeared crack model in Abaqus/Standard, described in “An inelastic constitutive model for concrete,” Section 4.5.1, and the brittle cracking model in Abaqus/Explicit, described in “A cracking model for concrete and other brittle materials,” Section 4.5.3.The concrete damaged plasticity model is primarily intended to provide a general capability for the analysis of concrete structures under cyclic and/or dynamic loading. The model is also suitable for the analysis of other quasi-brittle materials, such as rock, mortar and ceramics; but it is the behavior of concrete that is used in the remainder of this section to motivate different aspects of the constitutive theory. Under low confining pressures, concrete behaves in a brittle manner; the main failure mechanisms are cracking in tension and crushing in compression. The brittle behavior of concrete disappears when the confining pressure is sufficiently large to prevent crack propagation. In these circumstances failure is driven by the consolidation and collapse of the concrete microporous microstructure, leading to a macroscopic response that resembles that of a ductile material with work hardening.Modeling the behavior of concrete under large hydrostatic pressures is out of the scope of the plastic-damage model considered here. The constitutive theory in this section aims to capture the effects of irreversible damage associated with the failure mechanisms that occur in concrete and other quasi-brittle materials under fairly low confining pressures (less than four or five times the ultimate compressive stress in uniaxial compression loading). These effects manifest themselves in the following macroscopic properties:different yield strengths in tension and compression, with the initial yield stress in compression being a factor of 10 or more higher than the initial yield stress in tension;softening behavior in tension as opposed to initial hardening followed by softening in compression;different degradation of the elastic stiffness in tension and compression;stiffness recovery effects during cyclic loading; andrate sensitivity, especially an increase in the peak strength with strain rate.The plastic-damage model in Abaqus is based on the models proposed by Lubliner et al. (1989) and by Lee and Fenves (1998). The model is described in the remainder of this section. An overview of the main ingredients of the model is given first, followed by a more detailed discussion of the different aspects of the constitutive model.OverviewThe main ingredients of the inviscid concrete damaged plasticity model are summarized below.Strain rate decompositionAn additive strain rate decomposition is assumed for the rate-independent model:where is the total strain rate, is the elastic part of the strain rate, and is the plastic part of the strain rate.Stress-strain relationsThe stress-strain relations are governed by scalar damaged elasticity:where is the initial (undamaged) elastic stiffness of the material; is the degraded elastic stiffness; and d is the scalar stiffness degradation variable, which can take values in the range from zero (undamaged material) to one (fully damaged material). Damage associated with the failure mechanisms of the concrete (cracking and crushing) therefore results in a reduction in the elastic stiffness. Within the context of the scalar-damage theory, the stiffness degradation is isotropic and characterized by a single degradation variable, d. Following the usual notions of continuum damage mechanics, the effective stress is defined asThe Cauchy stress is related to the effective stress through the scalar degradation relation:For any given cross-section of the material, the factor represents the ratio of the effective load-carrying area (i.e., the overall area minus the damaged area) to the overall section area. In the absence of damage, , the effective stress is equivalent to the Cauchy stress, . When damage occurs, however, the effective stress is more representative than the Cauchy stress because it is the effective stress area that is resisting the external loads. It is, therefore, convenient to formulate the plasticity problem in terms of the effective stress. As discussed later, the evolution of the degradation variable is governed by a set of hardening variables, , and the effective stress; that is, .Hardening variablesDamaged states in tension and compression are characterized independently by two hardening variables, and , which are referred to as equivalen