The model assumes that the uniaxial tensile and compressive response of concrete is characterized by damaged plasticity, as shown in Figure 11.5.3–1.

It is assumed that the uniaxial stress-strain curves can be converted into stress versus plastic-strain curves. (This conversion is performed automatically by ABAQUS from the user-provided stress versus “inelastic” strain data, as explained below.) Thus,

As shown in Figure 11.5.3–1, when the concrete specimen is unloaded from any point on the strain softening branch of the stress-strain curves, the unloading response is weakened: the elastic stiffness of the material appears to be damaged (or degraded). The degradation of the elastic stiffness is characterized by two damage variables, and , which are assumed to be functions of the plastic strains, temperature, and field variables:

If is the initial (undamaged) elastic stiffness of the material, the stress-strain relations under uniaxial tension and compression loading are, respectively:

Under uniaxial cyclic loading conditions the degradation mechanisms are quite complex, involving the opening and closing of previously formed micro-cracks, as well as their interaction. Experimentally, it is observed that there is some recovery of the elastic stiffness as the load changes sign during a uniaxial cyclic test. The stiffness recovery effect, also known as the “unilateral effect,” is an important aspect of the concrete behavior under cyclic loading. The effect is usually more pronounced as the load changes from tension to compression, causing tensile cracks to close, which results in the recovery of the compressive stiffness.

The concrete damaged plasticity model assumes that the reduction of the elastic modulus is given in terms of a scalar degradation variable as

This expression holds both in the tensile () and the compressive () sides of the cycle. The stiffness degradation variable, , is a function of the stress state and the uniaxial damage variables, and . For the uniaxial cyclic conditions ABAQUS assumes that

The stress-strain relations for the general three-dimensional multiaxial condition are given by the scalar damage elasticity equation:

The previous expression for the scalar stiffness degradation variable, , is generalized to the multiaxial stress case by replacing the unit step function with a multiaxial stress weight factor, , defined as

See “Damaged plasticity model for concrete and other quasi-brittle materials,” Section 4.5.2 of the ABAQUS Theory Manual, for further details of the constitutive model.

In reinforced concrete the specification of postfailure behavior generally means giving the postfailure stress as a function of cracking strain, . The cracking strain is defined as the total strain minus the elastic strain corresponding to the undamaged material; that is, , where , as illustrated in Figure 11.5.3–3.

Tension stiffening data are given in terms of the cracking strain, . When unloading data are available, the data are provided to ABAQUS in terms of tensile damage curves, , as discussed below. ABAQUS automatically converts the cracking strain values to plastic strain values using the relationship

In cases with little or no reinforcement, the specification of a postfailure stress-strain relation introduces mesh sensitivity in the results, in the sense that the finite element predictions do not converge to a unique solution as the mesh is refined because mesh refinement leads to narrower crack bands. This problem typically occurs if cracking failure occurs only at localized regions in the structure and mesh refinement does not result in the formation of additional cracks. If cracking failure is distributed evenly (either due to the effect of rebar or due to the presence of stabilizing elastic material, as in the case of plate bending), mesh sensitivity is less of a concern.

In practical calculations for reinforced concrete, the mesh is usually such that each element contains rebars. The interaction between the rebars and the concrete tends to reduce the mesh sensitivity, provided that a reasonable amount of tension stiffening is introduced in the concrete model to simulate this interaction. This requires an estimate of the tension stiffening effect, which depends on such factors as the density of reinforcement, the quality of the bond between the rebar and the concrete, the relative size of the concrete aggregate compared to the rebar diameter, and the mesh. A reasonable starting point for relatively heavily reinforced concrete modeled with a fairly detailed mesh is to assume that the strain softening after failure reduces the stress linearly to zero at a total strain of about 10 times the strain at failure. The strain at failure in standard concretes is typically 10^–4, which suggests that tension stiffening that reduces the stress to zero at a total strain of about 10^–3 is reasonable. This parameter should be calibrated to a particular case.

The choice of tension stiffening parameters is important since, generally, more tension stiffening makes it easier to obtain numerical solutions. Too little tension stiffening will cause the local cracking failure in the concrete to introduce temporarily unstable behavior in the overall response of the model. Few practical designs exhibit such behavior, so that the presence of this type of response in the analysis model usually indicates that the tension stiffening is unreasonably low.

*CONCRETE TENSION STIFFENING, TYPE=STRAIN (default)

When there is no reinforcement in significant regions of the model, the tension stiffening approach described above will introduce unreasonable mesh sensitivity into the results. However, it is generally accepted that Hillerborg's (1976) fracture energy proposal is adequate to allay the concern for many practical purposes. Hillerborg defines the energy required to open a unit area of crack, , as a material parameter, using brittle fracture concepts. With this approach the concrete's brittle behavior is characterized by a stress-displacement response rather than a stress-strain response. Under tension a concrete specimen will crack across some section. After it has been pulled apart sufficiently for most of the stress to be removed (so that the undamaged elastic strain is small), its length will be determined primarily by the opening at the crack. The opening does not depend on the specimen's length.

This fracture energy cracking model can be invoked by specifying the postfailure stress as a tabular function of cracking displacement, as shown in Figure 11.5.3–4.

Alternatively, the fracture energy, , can be specified directly as a material property; in this case, define the failure stress, , as a tabular function of the associated fracture energy. This model assumes a linear loss of strength after cracking, as shown in Figure 11.5.3–5.

If tensile damage, , is specified, ABAQUS automatically converts the cracking displacement values to “plastic” displacement values using the relationship

*CONCRETE TENSION STIFFENING, TYPE=DISPLACEMENT

*CONCRETE TENSION STIFFENING, TYPE=GFI

You can define the uniaxial tension damage variable, , as a tabular function of either cracking strain or cracking displacement.

*CONCRETE TENSION DAMAGE, TYPE=STRAIN (default)

*CONCRETE TENSION DAMAGE, TYPE=DISPLACEMENT

You can define the uniaxial compression damage variable, , as a tabular function of inelastic (crushing) strain.

*CONCRETE COMPRESSION DAMAGE

As discussed above, stiffness recovery is an important aspect of the mechanical response of concrete under cyclic loading. ABAQUS allows direct user specification of the stiffness recovery factors and .

The experimental observation in most quasi-brittle materials, including concrete, is that the compressive stiffness is recovered upon crack closure as the load changes from tension to compression. On the other hand, the tensile stiffness is not recovered as the load changes from compression to tension once crushing micro-cracks have developed. This behavior, which corresponds to and , is the default used by ABAQUS. Figure 11.5.3–7 illustrates a uniaxial load cycle assuming the default behavior.

Input File Usage:	Use the following option to specify the compression stiffness recovery factor, :
	*CONCRETE TENSION DAMAGE, COMPRESSION RECOVERY= Use the following option to specify the tension stiffness recovery factor, : *CONCRETE COMPRESSION DAMAGE, TENSION RECOVERY=

The effective stress is defined as

The concrete damaged plasticity model assumes nonassociated potential plastic flow. The flow potential G used for this model is the Drucker-Prager hyperbolic function:

The default flow potential eccentricity is , which implies that the material has almost the same dilation angle over a wide range of confining pressure stress values. Increasing the value of provides more curvature to the flow potential, implying that the dilation angle increases more rapidly as the confining pressure decreases. Values of that are significantly less than the default value may lead to convergence problems if the material is subjected to low confining pressures because of the very tight curvature of the flow potential locally where it intersects the -axis.

The model makes use of the yield function of Lubliner et. al. (1989), with the modifications proposed by Lee and Fenves (1998) to account for different evolution of strength under tension and compression. The evolution of the yield surface is controlled by the hardening variables, and . In terms of effective stresses, the yield function takes the form

Typical yield surfaces are shown in Figure 11.5.3–8 on the deviatoric plane and in Figure 11.5.3–9 for plane stress conditions.

Because plastic flow is nonassociated, the use of concrete damaged plasticity results in a nonsymmetric material stiffness matrix. Therefore, to obtain an acceptable rate of convergence in ABAQUS/Standard, the unsymmetric matrix storage and solution scheme should be used. ABAQUS/Standard will automatically activate the unsymmetric solution scheme if concrete damaged plasticity is used in the analysis. If desired, you can turn off the unsymmetric solution scheme for a particular step (see “Procedures: overview,” Section 6.1.1).

Material models exhibiting softening behavior and stiffness degradation often lead to severe convergence difficulties in implicit analysis programs, such as ABAQUS/Standard. A common technique to overcome some of these convergence difficulties is the use of a viscoplastic regularization of the constitutive equations, which causes the consistent tangent stiffness of the softening material to become positive for sufficiently small time increments.

The concrete damaged plasticity model can be regularized in ABAQUS/Standard using viscoplasticity by permitting stresses to be outside of the yield surface. We use a generalization of the Duvaut-Lions regularization, according to which the viscoplastic strain rate tensor, , is defined as

Similarly, a viscous stiffness degradation variable, , for the viscoplastic system is defined as

Using the viscoplastic regularization with a small value for the viscosity parameter (small compared to the characteristic time increment) usually helps improve the rate of convergence of the model in the softening regime, without compromising results. The basic idea is that the solution of the viscoplastic system relaxes to that of the inviscid case as , where represents time. You can specify the value of the viscosity parameter as part of the concrete damaged plasticity material behavior definition. If the viscosity parameter is different from zero, output results of the plastic strain and stiffness degradation refer to the viscoplastic values, and . In ABAQUS/Standard the default value of the viscosity parameter is zero, so that no viscoplastic regularization is performed.