Concrete Damaged Plasticity

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

Response of concrete to uniaxial loading in tension (a) and compression (b).

Under uniaxial tension the stress-strain response follows a linear elastic relationship until the value of the failure stress, ${\sigma }_{t0}$, is reached. The failure stress corresponds to the onset of micro-cracking in the concrete material. Beyond the failure stress the formation of micro-cracks is represented macroscopically with a softening stress-strain response, which induces strain localization in the concrete structure. Under uniaxial compression the response is linear until the value of initial yield, ${\sigma }_{c0}$. In the plastic regime the response is typically characterized by stress hardening followed by strain softening beyond the ultimate stress, ${\sigma }_{cu}$. This representation, although somewhat simplified, captures the main features of the response of concrete.

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,

where the subscripts t and c refer to tension and compression, respectively; ${\tilde{\epsilon }}_t^{pl}$ and ${\tilde{\epsilon }}_c^{pl}$ are the equivalent plastic strains, ${\dot{\tilde{\epsilon }}}_t^{pl}$ and ${\dot{\tilde{\epsilon }}}_c^{pl}$ are the equivalent plastic strain rates, $\theta$ is the temperature, and $f_i,(i=1,2,\mathrm{\dots })$ are other predefined field variables.

As shown in Figure 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, $d_t$ and $d_c$, which are assumed to be functions of the plastic strains, temperature, and field variables:

The damage variables can take values from zero, representing the undamaged material, to one, which represents total loss of strength.

If $E_0$ is the initial (undamaged) elastic stiffness of the material, the stress-strain relations under uniaxial tension and compression loading are, respectively:

We define the “effective” tensile and compressive cohesion stresses as

The effective cohesion stresses determine the size of the yield (or failure) surface.

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 d as

where $E_0$ is the initial (undamaged) modulus of the material.

This expression holds both in the tensile (${\sigma }_{11}>0$) and the compressive (${\sigma }_{11}<0$) sides of the cycle. The stiffness degradation variable, d, is a function of the stress state and the uniaxial damage variables, $d_t$ and $d_c$. For the uniaxial cyclic conditions Abaqus assumes that

where $s_t$ and $s_c$ are functions of the stress state that are introduced to model stiffness recovery effects associated with stress reversals. They are defined according to

where

The weight factors $w_t$ and $w_c$, which are assumed to be material properties, control the recovery of the tensile and compressive stiffness upon load reversal. To illustrate this, consider the example in Figure 2, where the load changes from tension to compression.

Illustration of the effect of the compression stiffness recovery parameter $w_c$.

Assume that there was no previous compressive damage (crushing) in the material; that is, ${\tilde{\epsilon }}_c^{pl}=0$ and $d_c=0$. Then

• In tension (${\sigma }_{11}>0$), $r^{*}=1$; therefore, $d=d_t$ as expected.

• In compression (${\sigma }_{11}<0$), $r^{*}=0$, and $d=\left(1-w_c\right)d_t$. If $w_c=1$, then $d=0$; therefore, the material fully recovers the compressive stiffness (which in this case is the initial undamaged stiffness, $E=E_0$). If, on the other hand, $w_c=0$, then $d=d_t$ and there is no stiffness recovery. Intermediate values of $w_c$ result in partial recovery of the stiffness.

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

where ${\mathbf{D}}_0^{el}$ is the initial (undamaged) elasticity matrix.

The previous expression for the scalar stiffness degradation variable, d, is generalized to the multiaxial stress case by replacing the unit step function $r^{*}\left({\sigma }_{11}\right)$ with a multiaxial stress weight factor, $r\left(\widehat{\mathit{\sigma }}\right)$, defined as

where ${\widehat{\sigma }}_i\mathit{ }(i=1,2,3)$ are the principal stress components. The Macauley bracket $⟨\cdot ⟩$ is defined by $⟨x⟩=\frac{1}{2}\left(\left|x\right|+x\right)$.

See Damaged plasticity model for concrete and other quasi-brittle materials 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, ${\tilde{\epsilon }}_t^{ck}$. The cracking strain is defined as the total strain minus the elastic strain corresponding to the undamaged material; that is, ${\tilde{\epsilon }}_t^{ck}={\epsilon }_t-{\epsilon }_{0t}^{el}$, where ${\epsilon }_{0t}^{el}={\sigma }_t/E_0$, as illustrated in Figure 3. To avoid potential numerical problems, Abaqus enforces a lower limit on the postfailure stress equal to one-hundreth of the initial failure stress: ${\sigma }_t\ge {\sigma }_{t0}/100$.

Illustration of the definition of the cracking strain ${\tilde{\epsilon }}_t^{ck}$ used for the definition of tension stiffening data.

Tension stiffening data are given in terms of the cracking strain, ${\tilde{\epsilon }}_t^{ck}$. When unloading data are available, the data are provided to Abaqus in terms of tensile damage curves, $d_t-{\tilde{\epsilon }}_t^{ck}$, as discussed below. Abaqus automatically converts the cracking strain values to plastic strain values using the relationship

Abaqus will issue an error message if the calculated plastic strain values are negative and/or decreasing with increasing cracking strain, which typically indicates that the tensile damage curves are incorrect. In the absence of tensile damage ${\tilde{\epsilon }}_t^{pl}={\tilde{\epsilon }}_t^{ck}$.

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.

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, $G_f$, 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 4.

Postfailure stress-displacement curve.

Alternatively, the fracture energy, $G_f$, can be specified directly as a material property; in this case, define the failure stress, ${\sigma }_{t0}$, as a tabular function of the associated fracture energy. This model assumes a linear loss of strength after cracking, as shown in Figure 5.

Postfailure stress-fracture energy curve.

The cracking displacement at which complete loss of strength takes place is, therefore, $u_{t0}=2G_f/{\sigma }_{t0}$. Typical values of $G_f$ range from 40 N/m (0.22 lb/in) for a typical construction concrete (with a compressive strength of approximately 20 MPa, 2850 lb/in²) to 120 N/m (0.67 lb/in) for a high-strength concrete (with a compressive strength of approximately 40 MPa, 5700 lb/in²).

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

where $l_0$ is the specimen length. By default, Abaqus assumes $l_0$ to be one unit length ( $l_0=1$ ). However, it is recommended that you specify this value directly in the material definition to properly account for the length units.

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

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

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 $w_t$ and $w_c$.

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 $w_t=0$ and $w_c=1$, is the default used by Abaqus. Figure 7 illustrates a uniaxial load cycle assuming the default behavior.

Uniaxial load cycle (tension-compression-tension) assuming default values for the stiffness recovery factors: $w_t=0$ and $w_c=1$.

The effective stress is defined as

The plastic flow potential function and the yield surface make use of two stress invariants of the effective stress tensor, namely the hydrostatic pressure stress,

and the Mises equivalent effective stress,

where $\overline{\mathbf{S}}$ is the effective stress deviator, 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:

where

$\psi \left(\theta ,f_i\right)$

is the dilation angle measured in the p–q plane at high confining pressure;

${\sigma }_{t0}\left(\theta ,f_i\right)={{\sigma }_t|}_{{\tilde{\epsilon }}_t^{pl}=0,{\dot{\tilde{\epsilon }}}_t^{pl}=0}$

is the uniaxial tensile stress at failure, taken from the user-specified tension stiffening data; and

$ϵ\left(\theta ,f_i\right)$

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 always uniquely defined. The function approaches the linear Drucker-Prager flow potential asymptotically at high confining pressure stress and intersects the hydrostatic pressure axis at 90°. See Models for granular or polymer behavior for further discussion of this potential.

The default flow potential eccentricity is $ϵ=0.1$, which implies that the material has almost the same dilation angle over a wide range of confining pressure stress values. Increasing the value of $ϵ$ provides more curvature to the flow potential, implying that the dilation angle increases more rapidly as the confining pressure decreases. Values of $ϵ$ 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 p-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, ${\tilde{\epsilon }}_t^{pl}$ and ${\tilde{\epsilon }}_c^{pl}$. In terms of effective stresses, the yield function takes the form

with

Here,

${\widehat{\overline{\sigma }}}_{\max }$

is the maximum principal effective stress;

${\sigma }_{b0}/{\sigma }_{c0}$

is the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress (the default value is $1.16$);

$K_c$

is the ratio of the second stress invariant on the tensile meridian, $q_{\left(\mathrm{TM}\right)}$, to that on the compressive meridian, $q_{\left(\mathrm{CM}\right)}$, at initial yield for any given value of the pressure invariant p such that the maximum principal stress is negative, ${\widehat{\sigma }}_{\max }<0$ (see Figure 8); it must satisfy the condition $0.5<K_c\le 1.0$ (the default value is $2/3$);

${\overline{\sigma }}_t\left({\tilde{\epsilon }}_t^{pl}\right)$

is the effective tensile cohesion stress; and

${\overline{\sigma }}_c\left({\tilde{\epsilon }}_c^{pl}\right)$

is the effective compressive cohesion stress.

Yield surfaces in the deviatoric plane, corresponding to different values of $K_c$.

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

Yield surface in plane stress.

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 Defining an Analysis).

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, ${\dot{\mathit{\epsilon }}}_v^{pl}$, is defined as

Here $\mu$ is the viscosity parameter representing the relaxation time of the viscoplastic system, and ${\mathit{\epsilon }}^{pl}$ is the plastic strain evaluated in the inviscid backbone model.

Similarly, a viscous stiffness degradation variable, $d_v$, 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 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 $t/\mu \to \mathrm{\infty }$, where t 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, ${\mathit{\epsilon }}_v^{pl}$ and $d_v$. In Abaqus/Standard the default value of the viscosity parameter is zero, so that no viscoplastic regularization is performed.