混凝土或其他脆性材料的塑性破坏

Damaged plasticity model for concrete and other quasi-brittle materialsProducts:Abaqus/Standard Abaqus/ExplicitThis 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 inAbaqus/Standard,described in “Aninelastic constitutive model for concrete,”Section 4.5.1, and the brittle cracking model in Abaqus/Explicit, described in “A cracking modelfor concrete and other brittle materials,” Section 4.5.3.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.Overviewwhere 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 variablesas described later in this section.Microcracking and crushing in the concrete are represented by increasing values of the hardening variables.These variables control the evolution of the yield surface and the degradation of the elastic stiffness.They are also intimately related to the dissipated fracture energy required to generate micro-cracks.Yield functionThe yield function, , represents a surface in effective stress space, which determines the states of failure or damage. For the inviscid plastic-damage modelThe specific form of the yield function is described later in this section.Flow rulePlastic flow is governed by a flow potential G according to the flow rule:where is the nonnegative plastic multiplier. The plastic potential is defined in the effective stress space.The specific form of the flow potential for the concrete damaged plasticity model is discussed later in this section.The model uses nonassociated plasticity,therefore requiring the solution of nonsymmetric equations.SummaryIn summary, the elastic-plastic response of the concrete damaged plasticity model is described in terms of the effective stress and the hardening variables:where and F obey the Kuhn-Tucker conditions:The Cauchy stress is calculated in terms of the stiffness degradation variable, , and the effective stress asThe constitutive relations for the elastic-plastic response,Equation 4.5.2–1, aredecoupled from the stiffness degradation response,Equation 4.5.2–2,which makes the model attractive for an effective numerical implementation. The inviscid model summarized here can be extended easily to account for viscoplastic effects through the use of a viscoplastic regularization by permitting stresses to be outside the yield surface.Damage and stiffness degradationThe evolution equations of the hardening variables and are conveniently formulated by considering uniaxial loading conditions first and then extended to multiaxial conditions.Uniaxial conditionsIt is assumed that the uniaxial stress-strain curves can be converted into stress versus plastic strain curves of the formUnder uniaxial loading conditions the effective plastic strain rates are given asAs shown in Figure 4.5.2–1, when the concrete specimen is unloaded from any point on the strainsoftening branch of the stress-strain curves, the unloading response is observed to be weakened:The effective uniaxial cohesion stresses determine the size of the yield (or failure) surface.Uniaxial cyclic conditionsThe concrete damaged plasticity model assumes that the reduction of the elastic modulus is given in terms of a scalar degradation variable, d, aswhere is the initial (undamaged) modulus of the material.where and are functions of the stress state that are introduced to represent stiffness recovery effects associated with stress reversals. They are defined according towhereThe evolution equations of the equivalent plastic strains are also generalized to the uniaxial cyclic conditions asThe evolution equations for the hardening variables must be extended for the general multiaxial conditions. Based on Lee and Fenves(1998) we assume that the equivalent plastic strain rates are evaluated according to the expressionswhere and are, respectively, the maximum and minimum eigenvalues of the plastic strain rate tensor andIf the eigenvalues of the plastic strain rate tensor () are ordered such that , the evolution equation for general multiaxial stress conditions can be expressed in the following matrix form:whereandElastic stiffness degradationThe plastic-damage concrete model assumes that the elastic stiffness degradation is isotropic and characterized by a single scalar variable, d:similar to the uniaxial cyclic case, only that and are now given in terms of the function as It can be easily verified that Equation 4.5.2–10for the scalar degradation variable is consistent with the uniaxial response.and .Yield conditionThe plastic-damage concrete model uses a yield condition based on the yield function proposed by Lubliner et al.(1989) and incorporates the modifications proposed by Lee and Fenveswhere and are dimensionless material constants;is the effective hydrostatic pressure;is the Mises equivalent effective stress;is the deviatoric part of the effective stress tensor ; and is the algebraically maximum eigenvalue of . The function is given asTypical experimental values of the ratio for concrete are in the range from 1.10 to 1.16, yielding values of between 0.08 and 0.12 (Lubliner et al., 1989).Let for any given value of the hydrostatic pressure with ; thenThe fact that is constant does not seem to be contradicted by experimental evidence (Lubliner et al., 1989). The coefficient is, therefore, evaluated asA value of , which is typical for concrete, givesLet for any given value of the hydrostatic pressure with ; thenTypical yield surfaces are shown in Figure 4.5.2–4 in the deviatoric plane and in Figure 4.5.2–5for plane-stress conditions.Figure 4.5.2–4 Yield surfaces in the deviatoric plane, corresponding to different values of .Figure 4.5.2–5 Yield surface in plane stress.Flow ruleThe plastic-damage model assumes nonassociated potential flow,The flow potential G chosen for this model is the Drucker-Prager hyperbolic function:where is the dilation angle measured in the p–q plane at high confing pressure;is the uniaxial tensile stress at failure; and is a parameter, referred to as the eccentricity, that defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero). This flow potential, which is continuous and smooth, ensures that the flow direction is defined uniquely.The function asymptotically approaches the linear Drucker-Prager flow potential at high confing pressure stress and intersects the hydrostatic pressure axis at 90°.See “Modelsfor granular or polymer behavior,”Section 4.4.2,for further discussion of this potential.Because plastic flow is nonassociated, the use of the plastic-damage concrete model requires the solution of nonsymmetric equations.Viscoplastic regularizationHere is the viscosity parameter representing the relaxation time of the viscoplastic system and is the plastic strain evaluated in the inviscid backbone model.Similarly, a viscous stiffness degradation variable, , for the viscoplastic system is defined as where d is the degradation variable evaluated in the inviscid backbone model.The stress-strain relation of the viscoplastic model is given asIntegration of the modelThe model is integrated using the backward Euler method generally used with the plasticity models in Abaqus.A material Jacobian consistent with this integration operator is used for the equilibrium iterations.。